\begin{patch}{ugIntroCalcDerivPagePatch1}
\begin{paste}{ugIntroCalcDerivPageFull1}{ugIntroCalcDerivPageEmpty1}
\pastebutton{ugIntroCalcDerivPageFull1}{\hidepaste}
\tab{5}\spadcommand{f := exp exp x\bound{f }}
\indentrel{3}\begin{verbatim}
            x
          %e
   (1)  %e
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty1}
\begin{paste}{ugIntroCalcDerivPageEmpty1}{ugIntroCalcDerivPagePatch1}
\pastebutton{ugIntroCalcDerivPageEmpty1}{\showpaste}
\tab{5}\spadcommand{f := exp exp x\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch2}
\begin{paste}{ugIntroCalcDerivPageFull2}{ugIntroCalcDerivPageEmpty2}
\pastebutton{ugIntroCalcDerivPageFull2}{\hidepaste}
\tab{5}\spadcommand{D(f, x)\free{f }}
\indentrel{3}\begin{verbatim}
               x
          x  %e
   (2)  %e %e
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty2}
\begin{paste}{ugIntroCalcDerivPageEmpty2}{ugIntroCalcDerivPagePatch2}
\pastebutton{ugIntroCalcDerivPageEmpty2}{\showpaste}
\tab{5}\spadcommand{D(f, x)\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch3}
\begin{paste}{ugIntroCalcDerivPageFull3}{ugIntroCalcDerivPageEmpty3}
\pastebutton{ugIntroCalcDerivPageFull3}{\hidepaste}
\tab{5}\spadcommand{D(f, x, 4)\free{f }}
\indentrel{3}\begin{verbatim}
                                                x
            x 4        x 3        x 2     x   %e
   (3)  ((%e )  + 6 (%e )  + 7 (%e )  + %e )%e
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty3}
\begin{paste}{ugIntroCalcDerivPageEmpty3}{ugIntroCalcDerivPagePatch3}
\pastebutton{ugIntroCalcDerivPageEmpty3}{\showpaste}
\tab{5}\spadcommand{D(f, x, 4)\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch4}
\begin{paste}{ugIntroCalcDerivPageFull4}{ugIntroCalcDerivPageEmpty4}
\pastebutton{ugIntroCalcDerivPageFull4}{\hidepaste}
\tab{5}\spadcommand{g := sin(x^2 + y)\bound{g }}
\indentrel{3}\begin{verbatim}
                 2
   (4)  sin(y + x )
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty4}
\begin{paste}{ugIntroCalcDerivPageEmpty4}{ugIntroCalcDerivPagePatch4}
\pastebutton{ugIntroCalcDerivPageEmpty4}{\showpaste}
\tab{5}\spadcommand{g := sin(x^2 + y)\bound{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch5}
\begin{paste}{ugIntroCalcDerivPageFull5}{ugIntroCalcDerivPageEmpty5}
\pastebutton{ugIntroCalcDerivPageFull5}{\hidepaste}
\tab{5}\spadcommand{D(g, y)\free{g }}
\indentrel{3}\begin{verbatim}
                 2
   (5)  cos(y + x )
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty5}
\begin{paste}{ugIntroCalcDerivPageEmpty5}{ugIntroCalcDerivPagePatch5}
\pastebutton{ugIntroCalcDerivPageEmpty5}{\showpaste}
\tab{5}\spadcommand{D(g, y)\free{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch6}
\begin{paste}{ugIntroCalcDerivPageFull6}{ugIntroCalcDerivPageEmpty6}
\pastebutton{ugIntroCalcDerivPageFull6}{\hidepaste}
\tab{5}\spadcommand{D(g, [y, y, x, x])\free{g }}
\indentrel{3}\begin{verbatim}
           2         2               2
   (6)  4 x sin(y + x ) - 2 cos(y + x )
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty6}
\begin{paste}{ugIntroCalcDerivPageEmpty6}{ugIntroCalcDerivPagePatch6}
\pastebutton{ugIntroCalcDerivPageEmpty6}{\showpaste}
\tab{5}\spadcommand{D(g, [y, y, x, x])\free{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch7}
\begin{paste}{ugIntroCalcDerivPageFull7}{ugIntroCalcDerivPageEmpty7}
\pastebutton{ugIntroCalcDerivPageFull7}{\hidepaste}
\tab{5}\spadcommand{D(F,x)}
\indentrel{3}\begin{verbatim}
   (7)  0
                              Type: Polynomial(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty7}
\begin{paste}{ugIntroCalcDerivPageEmpty7}{ugIntroCalcDerivPagePatch7}
\pastebutton{ugIntroCalcDerivPageEmpty7}{\showpaste}
\tab{5}\spadcommand{D(F,x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch8}
\begin{paste}{ugIntroCalcDerivPageFull8}{ugIntroCalcDerivPageEmpty8}
\pastebutton{ugIntroCalcDerivPageFull8}{\hidepaste}
\tab{5}\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y\bound{F x y }}
\indentrel{3}\begin{verbatim}
   (8)  y
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty8}
\begin{paste}{ugIntroCalcDerivPageEmpty8}{ugIntroCalcDerivPagePatch8}
\pastebutton{ugIntroCalcDerivPageEmpty8}{\showpaste}
\tab{5}\spadcommand{F := operator 'F; x := operator 'x; y := operator 'y\bound{F x y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch9}
\begin{paste}{ugIntroCalcDerivPageFull9}{ugIntroCalcDerivPageEmpty9}
\pastebutton{ugIntroCalcDerivPageFull9}{\hidepaste}
\tab{5}\spadcommand{a := F(x z, y z, z^2) + x y(z+1)\bound{a }\free{F }\free{x }\free{y }}
\indentrel{3}\begin{verbatim}
                                   2
   (9)  x(y(z + 1)) + F(x(z),y(z),z )
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty9}
\begin{paste}{ugIntroCalcDerivPageEmpty9}{ugIntroCalcDerivPagePatch9}
\pastebutton{ugIntroCalcDerivPageEmpty9}{\showpaste}
\tab{5}\spadcommand{a := F(x z, y z, z^2) + x y(z+1)\bound{a }\free{F }\free{x }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch10}
\begin{paste}{ugIntroCalcDerivPageFull10}{ugIntroCalcDerivPageEmpty10}
\pastebutton{ugIntroCalcDerivPageFull10}{\hidepaste}
\tab{5}\spadcommand{dadz := D(a, z)\bound{da }\free{a }}
\indentrel{3}\begin{verbatim}
   (10)
                       2     ,                  2
     2 zF  (x(z),y(z),z ) + y (z)F  (x(z),y(z),z )
         ,3                       ,2
   + 
      ,                  2     ,           ,
     x (z)F  (x(z),y(z),z ) + x (y(z + 1))y (z + 1)
           ,1
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty10}
\begin{paste}{ugIntroCalcDerivPageEmpty10}{ugIntroCalcDerivPagePatch10}
\pastebutton{ugIntroCalcDerivPageEmpty10}{\showpaste}
\tab{5}\spadcommand{dadz := D(a, z)\bound{da }\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch11}
\begin{paste}{ugIntroCalcDerivPageFull11}{ugIntroCalcDerivPageEmpty11}
\pastebutton{ugIntroCalcDerivPageFull11}{\hidepaste}
\tab{5}\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{da }}
\indentrel{3}\begin{verbatim}
   (11)
           2             z             2
       (2 z  + 2 z)F  (%e ,log(z + 1),z )
                    ,3
     + 
             z             2
       F  (%e ,log(z + 1),z )
        ,2
     + 
                z      z             2
       (z + 1)%e F  (%e ,log(z + 1),z ) + z + 1
                  ,1
  /
     z + 1
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty11}
\begin{paste}{ugIntroCalcDerivPageEmpty11}{ugIntroCalcDerivPagePatch11}
\pastebutton{ugIntroCalcDerivPageEmpty11}{\showpaste}
\tab{5}\spadcommand{eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{da }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch12}
\begin{paste}{ugIntroCalcDerivPageFull12}{ugIntroCalcDerivPageEmpty12}
\pastebutton{ugIntroCalcDerivPageFull12}{\hidepaste}
\tab{5}\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{a }\bound{eva }}
\indentrel{3}\begin{verbatim}
             z             2
   (12)  F(%e ,log(z + 1),z ) + z + 2
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty12}
\begin{paste}{ugIntroCalcDerivPageEmpty12}{ugIntroCalcDerivPagePatch12}
\pastebutton{ugIntroCalcDerivPageEmpty12}{\showpaste}
\tab{5}\spadcommand{eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))\free{a }\bound{eva }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPagePatch13}
\begin{paste}{ugIntroCalcDerivPageFull13}{ugIntroCalcDerivPageEmpty13}
\pastebutton{ugIntroCalcDerivPageFull13}{\hidepaste}
\tab{5}\spadcommand{D(\%, z)\free{eva }}
\indentrel{3}\begin{verbatim}
   (13)
           2             z             2
       (2 z  + 2 z)F  (%e ,log(z + 1),z )
                    ,3
     + 
             z             2
       F  (%e ,log(z + 1),z )
        ,2
     + 
                z      z             2
       (z + 1)%e F  (%e ,log(z + 1),z ) + z + 1
                  ,1
  /
     z + 1
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcDerivPageEmpty13}
\begin{paste}{ugIntroCalcDerivPageEmpty13}{ugIntroCalcDerivPagePatch13}
\pastebutton{ugIntroCalcDerivPageEmpty13}{\showpaste}
\tab{5}\spadcommand{D(\%, z)\free{eva }}
\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPagePatch1}
\begin{paste}{ugIntroVariablesPageFull1}{ugIntroVariablesPageEmpty1}
\pastebutton{ugIntroVariablesPageFull1}{\hidepaste}
\tab{5}\spadcommand{(x^2 - x*y^3 +3*y)^2}
\indentrel{3}\begin{verbatim}
         2 6        4      3 3      2      2     4
   (1)  x y  - 6 x y  - 2 x y  + 9 y  + 6 x y + x
                              Type: Polynomial(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPageEmpty1}
\begin{paste}{ugIntroVariablesPageEmpty1}{ugIntroVariablesPagePatch1}
\pastebutton{ugIntroVariablesPageEmpty1}{\showpaste}
\tab{5}\spadcommand{(x^2 - x*y^3 +3*y)^2}
\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPagePatch2}
\begin{paste}{ugIntroVariablesPageFull2}{ugIntroVariablesPageEmpty2}
\pastebutton{ugIntroVariablesPageFull2}{\hidepaste}
\tab{5}\spadcommand{p: UP(x,INT) := (3*x-1)^2 * (2*x + 8)}
\indentrel{3}\begin{verbatim}
            3       2
   (2)  18 x  + 60 x  - 46 x + 8
                  Type: UnivariatePolynomial(x,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPageEmpty2}
\begin{paste}{ugIntroVariablesPageEmpty2}{ugIntroVariablesPagePatch2}
\pastebutton{ugIntroVariablesPageEmpty2}{\showpaste}
\tab{5}\spadcommand{p: UP(x,INT) := (3*x-1)^2 * (2*x + 8)}
\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPagePatch3}
\begin{paste}{ugIntroVariablesPageFull3}{ugIntroVariablesPageEmpty3}
\pastebutton{ugIntroVariablesPageFull3}{\hidepaste}
\tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x^2-x*y^3+3*y)^2\bound{m }}
\indentrel{3}\begin{verbatim}
         4      3 3     6        2      4       2
   (3)  x  - 2 y x  + (y  + 6 y)x  - 6 y x + 9 y
            Type: MultivariatePolynomial([x,y],Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPageEmpty3}
\begin{paste}{ugIntroVariablesPageEmpty3}{ugIntroVariablesPagePatch3}
\pastebutton{ugIntroVariablesPageEmpty3}{\showpaste}
\tab{5}\spadcommand{m: MPOLY([x,y],INT) := (x^2-x*y^3+3*y)^2\bound{m }}
\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPagePatch4}
\begin{paste}{ugIntroVariablesPageFull4}{ugIntroVariablesPageEmpty4}
\pastebutton{ugIntroVariablesPageFull4}{\hidepaste}
\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }}
\indentrel{3}\begin{verbatim}
         2 6        4      3 3      2      2     4
   (4)  x y  - 6 x y  - 2 x y  + 9 y  + 6 x y + x
            Type: MultivariatePolynomial([y,x],Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPageEmpty4}
\begin{paste}{ugIntroVariablesPageEmpty4}{ugIntroVariablesPagePatch4}
\pastebutton{ugIntroVariablesPageEmpty4}{\showpaste}
\tab{5}\spadcommand{m :: MPOLY([y,x],INT)\free{m }}
\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPagePatch5}
\begin{paste}{ugIntroVariablesPageFull5}{ugIntroVariablesPageEmpty5}
\pastebutton{ugIntroVariablesPageFull5}{\hidepaste}
\tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }}
\indentrel{3}\begin{verbatim}
         6 2      4       3 3      2        2    4
   (5)  y x  - 6 y x - 2 y x  + 9 y  + 6 y x  + x
 Type: DistributedMultivariatePolynomial([y,x],Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPageEmpty5}
\begin{paste}{ugIntroVariablesPageEmpty5}{ugIntroVariablesPagePatch5}
\pastebutton{ugIntroVariablesPageEmpty5}{\showpaste}
\tab{5}\spadcommand{m :: DMP([y,x],INT)\free{m }}
\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPagePatch6}
\begin{paste}{ugIntroVariablesPageFull6}{ugIntroVariablesPageEmpty6}
\pastebutton{ugIntroVariablesPageFull6}{\hidepaste}
\tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }}
\indentrel{3}\begin{verbatim}
         6 2      3 3      4     4        2      2
   (6)  y x  - 2 y x  - 6 y x + x  + 6 y x  + 9 y
Type: HomogeneousDistributedMultivariatePolynomial([y,x],Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroVariablesPageEmpty6}
\begin{paste}{ugIntroVariablesPageEmpty6}{ugIntroVariablesPagePatch6}
\pastebutton{ugIntroVariablesPageEmpty6}{\showpaste}
\tab{5}\spadcommand{m :: HDMP([y,x],INT)\free{m }}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch1}
\begin{paste}{ugIntroIntegratePageFull1}{ugIntroIntegratePageEmpty1}
\pastebutton{ugIntroIntegratePageFull1}{\hidepaste}
\tab{5}\spadcommand{integrate((x^2+2*x+1)/((x+1)^6+1),x)}
\indentrel{3}\begin{verbatim}
              3      2
        atan(x  + 3 x  + 3 x + 1)
   (1)  -------------------------
                    3
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty1}
\begin{paste}{ugIntroIntegratePageEmpty1}{ugIntroIntegratePagePatch1}
\pastebutton{ugIntroIntegratePageEmpty1}{\showpaste}
\tab{5}\spadcommand{integrate((x^2+2*x+1)/((x+1)^6+1),x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch2}
\begin{paste}{ugIntroIntegratePageFull2}{ugIntroIntegratePageEmpty2}
\pastebutton{ugIntroIntegratePageFull2}{\hidepaste}
\tab{5}\spadcommand{integrate(1/(x^2 + a),x)}
\indentrel{3}\begin{verbatim}
               2      +---+
             (x  - a)\|- a  + 2 a x          +-+
         log(----------------------)       x\|a
                      2               atan(-----)
                     x  + a                  a
   (2)  [---------------------------, -----------]
                      +---+                +-+
                   2 \|- a                \|a
             Type: Union(List(Expression(Integer)),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty2}
\begin{paste}{ugIntroIntegratePageEmpty2}{ugIntroIntegratePagePatch2}
\pastebutton{ugIntroIntegratePageEmpty2}{\showpaste}
\tab{5}\spadcommand{integrate(1/(x^2 + a),x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch3}
\begin{paste}{ugIntroIntegratePageFull3}{ugIntroIntegratePageEmpty3}
\pastebutton{ugIntroIntegratePageFull3}{\hidepaste}
\tab{5}\spadcommand{complexIntegrate(1/(x^2 + a),x)}
\indentrel{3}\begin{verbatim}
   (3)
    +---+      +---+         +---+        +---+
    |  1       |  1          |  1         |  1
    |- - log(a |- -  + x) -  |- - log(- a |- -  + x)
   \|  a      \|  a         \|  a        \|  a
   -------------------------------------------------
                           2
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty3}
\begin{paste}{ugIntroIntegratePageEmpty3}{ugIntroIntegratePagePatch3}
\pastebutton{ugIntroIntegratePageEmpty3}{\showpaste}
\tab{5}\spadcommand{complexIntegrate(1/(x^2 + a),x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch4}
\begin{paste}{ugIntroIntegratePageFull4}{ugIntroIntegratePageEmpty4}
\pastebutton{ugIntroIntegratePageFull4}{\hidepaste}
\tab{5}\spadcommand{integrate(x^3 / (a+b*x)^(1/3),x)}
\indentrel{3}\begin{verbatim}
   (4)
         3 3          2 2        2           3 3+-------+2
   (120 b x  - 135 a b x  + 162 a b x - 243 a )\|b x + a
   -------------------------------------------------------
                                 4
                            440 b
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty4}
\begin{paste}{ugIntroIntegratePageEmpty4}{ugIntroIntegratePagePatch4}
\pastebutton{ugIntroIntegratePageEmpty4}{\showpaste}
\tab{5}\spadcommand{integrate(x^3 / (a+b*x)^(1/3),x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch5}
\begin{paste}{ugIntroIntegratePageFull5}{ugIntroIntegratePageEmpty5}
\pastebutton{ugIntroIntegratePageFull5}{\hidepaste}
\tab{5}\spadcommand{integrate(1 / (x^3 * (a+b*x)^(1/3)),x)}
\indentrel{3}\begin{verbatim}
   (5)
       -
               2 2 +-+
            2 b x \|3
         *
                3+-+3+-------+2   3+-+2 3+-------+
            log(\|a \|b x + a   + \|a   \|b x + a  + a)
     + 
          2 2 +-+    3+-+2 3+-------+
       4 b x \|3 log(\|a   \|b x + a  - a)
     + 
                      +-+3+-+2 3+-------+     +-+
           2 2     2 \|3 \|a   \|b x + a  + a\|3
       12 b x atan(------------------------------)
                                 3 a
     + 
                      +-+3+-+3+-------+2
       (12 b x - 9 a)\|3 \|a \|b x + a
  /
         2 2 +-+3+-+
     18 a x \|3 \|a
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty5}
\begin{paste}{ugIntroIntegratePageEmpty5}{ugIntroIntegratePagePatch5}
\pastebutton{ugIntroIntegratePageEmpty5}{\showpaste}
\tab{5}\spadcommand{integrate(1 / (x^3 * (a+b*x)^(1/3)),x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch6}
\begin{paste}{ugIntroIntegratePageFull6}{ugIntroIntegratePageEmpty6}
\pastebutton{ugIntroIntegratePageFull6}{\hidepaste}
\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
\indentrel{3}\begin{verbatim}
           x      +--------+
         ++  log(\|b + %Y a  + 1)
   (6)   |   -------------------- d%Y
        ++            %Y
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty6}
\begin{paste}{ugIntroIntegratePageEmpty6}{ugIntroIntegratePagePatch6}
\pastebutton{ugIntroIntegratePageEmpty6}{\showpaste}
\tab{5}\spadcommand{integrate(log(1 + sqrt(a*x + b)) / x,x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch7}
\begin{paste}{ugIntroIntegratePageFull7}{ugIntroIntegratePageEmpty7}
\pastebutton{ugIntroIntegratePageFull7}{\hidepaste}
\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)}
\indentrel{3}\begin{verbatim}
   (7)
                           +-----+
                 - 2 cosh(\|x + b  + 1) - 2 x
     2 log(---------------------------------------)
                 +-----+              +-----+
           sinh(\|x + b  + 1) - cosh(\|x + b  + 1)
   + 
          +-----+
     - 2 \|x + b
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty7}
\begin{paste}{ugIntroIntegratePageEmpty7}{ugIntroIntegratePagePatch7}
\pastebutton{ugIntroIntegratePageEmpty7}{\showpaste}
\tab{5}\spadcommand{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b)) / (sqrt(x+b) * (x + cosh(1+sqrt(x + b)))), x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch8}
\begin{paste}{ugIntroIntegratePageFull8}{ugIntroIntegratePageEmpty8}
\pastebutton{ugIntroIntegratePageFull8}{\hidepaste}
\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
\indentrel{3}\begin{verbatim}
   (8)
                   atan(x) 2              atan(x) 2
       8 log(3 tan(-------)  - 1) - 3 tan(-------)
                      3                      3
     + 
                atan(x)
       18 x tan(-------)
                   3
  /
     18
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty8}
\begin{paste}{ugIntroIntegratePageEmpty8}{ugIntroIntegratePagePatch8}
\pastebutton{ugIntroIntegratePageEmpty8}{\showpaste}
\tab{5}\spadcommand{integrate(tan(atan(x)/3),x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch9}
\begin{paste}{ugIntroIntegratePageFull9}{ugIntroIntegratePageEmpty9}
\pastebutton{ugIntroIntegratePageFull9}{\hidepaste}
\tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ^ (3/2)), x)}
\indentrel{3}\begin{verbatim}
             +----------+
          2 \|log(x) + x
   (9)  - ---------------
             log(x) + x
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty9}
\begin{paste}{ugIntroIntegratePageEmpty9}{ugIntroIntegratePagePatch9}
\pastebutton{ugIntroIntegratePageEmpty9}{\showpaste}
\tab{5}\spadcommand{integrate((x + 1) / (x*(x + log x) ^ (3/2)), x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePagePatch10}
\begin{paste}{ugIntroIntegratePageFull10}{ugIntroIntegratePageEmpty10}
\pastebutton{ugIntroIntegratePageFull10}{\hidepaste}
\tab{5}\spadcommand{integrate(exp(-x^2) * erf(x) / (erf(x)^3 - erf(x)^2 - erf(x) + 1),x)}
\indentrel{3}\begin{verbatim}
                      +---+    erf(x) - 1       +---+
         (erf(x) - 1)\|%pi log(----------) - 2 \|%pi
                               erf(x) + 1
   (10)  --------------------------------------------
                         8 erf(x) - 8
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroIntegratePageEmpty10}
\begin{paste}{ugIntroIntegratePageEmpty10}{ugIntroIntegratePagePatch10}
\pastebutton{ugIntroIntegratePageEmpty10}{\showpaste}
\tab{5}\spadcommand{integrate(exp(-x^2) * erf(x) / (erf(x)^3 - erf(x)^2 - erf(x) + 1),x)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPagePatch1}
\begin{paste}{ugIntroCalcLimitsPageFull1}{ugIntroCalcLimitsPageEmpty1}
\pastebutton{ugIntroCalcLimitsPageFull1}{\hidepaste}
\tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }}
\indentrel{3}\begin{verbatim}
         csc(a x)
   (1)  ---------
        csch(b x)
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPageEmpty1}
\begin{paste}{ugIntroCalcLimitsPageEmpty1}{ugIntroCalcLimitsPagePatch1}
\pastebutton{ugIntroCalcLimitsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{g := csc(a*x) / csch(b*x)\bound{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPagePatch2}
\begin{paste}{ugIntroCalcLimitsPageFull2}{ugIntroCalcLimitsPageEmpty2}
\pastebutton{ugIntroCalcLimitsPageFull2}{\hidepaste}
\tab{5}\spadcommand{limit(g,x=0)\free{g }}
\indentrel{3}\begin{verbatim}
        b
   (2)  -
        a
Type: Union(OrderedCompletion(Expression(Integer)),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPageEmpty2}
\begin{paste}{ugIntroCalcLimitsPageEmpty2}{ugIntroCalcLimitsPagePatch2}
\pastebutton{ugIntroCalcLimitsPageEmpty2}{\showpaste}
\tab{5}\spadcommand{limit(g,x=0)\free{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPagePatch3}
\begin{paste}{ugIntroCalcLimitsPageFull3}{ugIntroCalcLimitsPageEmpty3}
\pastebutton{ugIntroCalcLimitsPageFull3}{\hidepaste}
\tab{5}\spadcommand{h := (1 + k/x)^x\bound{h }}
\indentrel{3}\begin{verbatim}
         x + k x
   (3)  (-----)
           x
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPageEmpty3}
\begin{paste}{ugIntroCalcLimitsPageEmpty3}{ugIntroCalcLimitsPagePatch3}
\pastebutton{ugIntroCalcLimitsPageEmpty3}{\showpaste}
\tab{5}\spadcommand{h := (1 + k/x)^x\bound{h }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPagePatch4}
\begin{paste}{ugIntroCalcLimitsPageFull4}{ugIntroCalcLimitsPageEmpty4}
\pastebutton{ugIntroCalcLimitsPageFull4}{\hidepaste}
\tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }}
\indentrel{3}\begin{verbatim}
          k
   (4)  %e
Type: Union(OrderedCompletion(Expression(Integer)),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPageEmpty4}
\begin{paste}{ugIntroCalcLimitsPageEmpty4}{ugIntroCalcLimitsPagePatch4}
\pastebutton{ugIntroCalcLimitsPageEmpty4}{\showpaste}
\tab{5}\spadcommand{limit(h,x=\%plusInfinity)\free{h }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPagePatch5}
\begin{paste}{ugIntroCalcLimitsPageFull5}{ugIntroCalcLimitsPageEmpty5}
\pastebutton{ugIntroCalcLimitsPageFull5}{\hidepaste}
\tab{5}\spadcommand{limit(sqrt(y^2)/y,y = 0)}
\indentrel{3}\begin{verbatim}
   (5)  [leftHandLimit = - 1, rightHandLimit = 1]
Type: Union(Record(leftHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed")),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPageEmpty5}
\begin{paste}{ugIntroCalcLimitsPageEmpty5}{ugIntroCalcLimitsPagePatch5}
\pastebutton{ugIntroCalcLimitsPageEmpty5}{\showpaste}
\tab{5}\spadcommand{limit(sqrt(y^2)/y,y = 0)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPagePatch6}
\begin{paste}{ugIntroCalcLimitsPageFull6}{ugIntroCalcLimitsPageEmpty6}
\pastebutton{ugIntroCalcLimitsPageFull6}{\hidepaste}
\tab{5}\spadcommand{limit(exp(-1/x^2),x = 0)}
\indentrel{3}\begin{verbatim}
   (6)  0
Type: Union(OrderedCompletion(Expression(Integer)),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPageEmpty6}
\begin{paste}{ugIntroCalcLimitsPageEmpty6}{ugIntroCalcLimitsPagePatch6}
\pastebutton{ugIntroCalcLimitsPageEmpty6}{\showpaste}
\tab{5}\spadcommand{limit(exp(-1/x^2),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPagePatch7}
\begin{paste}{ugIntroCalcLimitsPageFull7}{ugIntroCalcLimitsPageEmpty7}
\pastebutton{ugIntroCalcLimitsPageFull7}{\hidepaste}
\tab{5}\spadcommand{complexLimit(exp(-1/x^2),x = 0)}
\indentrel{3}\begin{verbatim}
   (7)  "failed"
                              Type: Union("failed",...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCalcLimitsPageEmpty7}
\begin{paste}{ugIntroCalcLimitsPageEmpty7}{ugIntroCalcLimitsPagePatch7}
\pastebutton{ugIntroCalcLimitsPageEmpty7}{\showpaste}
\tab{5}\spadcommand{complexLimit(exp(-1/x^2),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch1}
\begin{paste}{ugIntroSysCmmandsPageFull1}{ugIntroSysCmmandsPageEmpty1}
\pastebutton{ugIntroSysCmmandsPageFull1}{\hidepaste}
\tab{5}\spadcommand{)what operations integrate}
\indentrel{3}\begin{verbatim}
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty1}
\begin{paste}{ugIntroSysCmmandsPageEmpty1}{ugIntroSysCmmandsPagePatch1}
\pastebutton{ugIntroSysCmmandsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{)what operations integrate}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch2}
\begin{paste}{ugIntroSysCmmandsPageFull2}{ugIntroSysCmmandsPageEmpty2}
\pastebutton{ugIntroSysCmmandsPageFull2}{\hidepaste}
\tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty2}
\begin{paste}{ugIntroSysCmmandsPageEmpty2}{ugIntroSysCmmandsPagePatch2}
\pastebutton{ugIntroSysCmmandsPageEmpty2}{\showpaste}
\tab{5}\spadcommand{f(0) == 1; g(0) == 1\bound{u1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch3}
\begin{paste}{ugIntroSysCmmandsPageFull3}{ugIntroSysCmmandsPageEmpty3}
\pastebutton{ugIntroSysCmmandsPageFull3}{\hidepaste}
\tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty3}
\begin{paste}{ugIntroSysCmmandsPageEmpty3}{ugIntroSysCmmandsPagePatch3}
\pastebutton{ugIntroSysCmmandsPageEmpty3}{\showpaste}
\tab{5}\spadcommand{f(n) == e/2*f(n-1) - x*g(n-1)\bound{u2 }\free{u1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch4}
\begin{paste}{ugIntroSysCmmandsPageFull4}{ugIntroSysCmmandsPageEmpty4}
\pastebutton{ugIntroSysCmmandsPageFull4}{\hidepaste}
\tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty4}
\begin{paste}{ugIntroSysCmmandsPageEmpty4}{ugIntroSysCmmandsPagePatch4}
\pastebutton{ugIntroSysCmmandsPageEmpty4}{\showpaste}
\tab{5}\spadcommand{g(n) == -x*f(n-1) + d/3*g(n-1)\bound{u3 }\free{u2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch5}
\begin{paste}{ugIntroSysCmmandsPageFull5}{ugIntroSysCmmandsPageEmpty5}
\pastebutton{ugIntroSysCmmandsPageFull5}{\hidepaste}
\tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }}
\indentrel{3}\begin{verbatim}
   (4)
      3        1    2      1  2   1       1  2     1  3
   - x  + (e + - d)x  + (- - e  - - d e - - d )x + - e
               3           4      6       9        8
                    Type: Polynomial(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty5}
\begin{paste}{ugIntroSysCmmandsPageEmpty5}{ugIntroSysCmmandsPagePatch5}
\pastebutton{ugIntroSysCmmandsPageEmpty5}{\showpaste}
\tab{5}\spadcommand{f(3)\bound{u4 }\free{u3 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch6}
\begin{paste}{ugIntroSysCmmandsPageFull6}{ugIntroSysCmmandsPageEmpty6}
\pastebutton{ugIntroSysCmmandsPageFull6}{\hidepaste}
\tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }}
\indentrel{3}\begin{verbatim}
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty6}
\begin{paste}{ugIntroSysCmmandsPageEmpty6}{ugIntroSysCmmandsPagePatch6}
\pastebutton{ugIntroSysCmmandsPageEmpty6}{\showpaste}
\tab{5}\spadcommand{)undo 2\bound{u5 }\free{u4 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch7}
\begin{paste}{ugIntroSysCmmandsPageFull7}{ugIntroSysCmmandsPageEmpty7}
\pastebutton{ugIntroSysCmmandsPageFull7}{\hidepaste}
\tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty7}
\begin{paste}{ugIntroSysCmmandsPageEmpty7}{ugIntroSysCmmandsPagePatch7}
\pastebutton{ugIntroSysCmmandsPageEmpty7}{\showpaste}
\tab{5}\spadcommand{f(n) == d/3*f(n-1) - x*g(n-1)\bound{u6 }\free{u5 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch8}
\begin{paste}{ugIntroSysCmmandsPageFull8}{ugIntroSysCmmandsPageEmpty8}
\pastebutton{ugIntroSysCmmandsPageFull8}{\hidepaste}
\tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }}
\indentrel{3}\begin{verbatim}
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty8}
\begin{paste}{ugIntroSysCmmandsPageEmpty8}{ugIntroSysCmmandsPagePatch8}
\pastebutton{ugIntroSysCmmandsPageEmpty8}{\showpaste}
\tab{5}\spadcommand{)undo )redo\bound{u7 }\free{u6 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch9}
\begin{paste}{ugIntroSysCmmandsPageFull9}{ugIntroSysCmmandsPageEmpty9}
\pastebutton{ugIntroSysCmmandsPageFull9}{\hidepaste}
\tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }}
\indentrel{3}\begin{verbatim}
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty9}
\begin{paste}{ugIntroSysCmmandsPageEmpty9}{ugIntroSysCmmandsPagePatch9}
\pastebutton{ugIntroSysCmmandsPageEmpty9}{\showpaste}
\tab{5}\spadcommand{)undo 4\bound{u8 }\free{u7 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPagePatch10}
\begin{paste}{ugIntroSysCmmandsPageFull10}{ugIntroSysCmmandsPageEmpty10}
\pastebutton{ugIntroSysCmmandsPageFull10}{\hidepaste}
\tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }}
\indentrel{3}\begin{verbatim}
   (6)
      3        1    2      1  2   1       1  2     1  3
   - x  + (e + - d)x  + (- - e  - - d e - - d )x + - e
               3           4      6       9        8
                    Type: Polynomial(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSysCmmandsPageEmpty10}
\begin{paste}{ugIntroSysCmmandsPageEmpty10}{ugIntroSysCmmandsPagePatch10}
\pastebutton{ugIntroSysCmmandsPageEmpty10}{\showpaste}
\tab{5}\spadcommand{f(3)\bound{u9 }\free{u8 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroArithmeticPagePatch1}
\begin{paste}{ugIntroArithmeticPageFull1}{ugIntroArithmeticPageEmpty1}
\pastebutton{ugIntroArithmeticPageFull1}{\hidepaste}
\tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ^ 2 - 1}
\indentrel{3}\begin{verbatim}
          19
   (1)  - --
           4
                                Type: Fraction(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroArithmeticPageEmpty1}
\begin{paste}{ugIntroArithmeticPageEmpty1}{ugIntroArithmeticPagePatch1}
\pastebutton{ugIntroArithmeticPageEmpty1}{\showpaste}
\tab{5}\spadcommand{1 + 2 - 3 / 4 * 3 ^ 2 - 1}
\end{paste}\end{patch}

\begin{patch}{ugIntroArithmeticPagePatch2}
\begin{paste}{ugIntroArithmeticPageFull2}{ugIntroArithmeticPageEmpty2}
\pastebutton{ugIntroArithmeticPageFull2}{\hidepaste}
\tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ^ 2))) - 1}
\indentrel{3}\begin{verbatim}
          19
   (2)  - --
           4
                                Type: Fraction(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroArithmeticPageEmpty2}
\begin{paste}{ugIntroArithmeticPageEmpty2}{ugIntroArithmeticPagePatch2}
\pastebutton{ugIntroArithmeticPageEmpty2}{\showpaste}
\tab{5}\spadcommand{((1 + 2) - ((3 / 4) * (3 ^ 2))) - 1}
\end{paste}\end{patch}

\begin{patch}{ugIntroArithmeticPagePatch3}
\begin{paste}{ugIntroArithmeticPageFull3}{ugIntroArithmeticPageEmpty3}
\pastebutton{ugIntroArithmeticPageFull3}{\hidepaste}
\tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ^ (2 - 1))}
\indentrel{3}\begin{verbatim}
        11
   (3)  --
         4
                                Type: Fraction(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroArithmeticPageEmpty3}
\begin{paste}{ugIntroArithmeticPageEmpty3}{ugIntroArithmeticPagePatch3}
\pastebutton{ugIntroArithmeticPageEmpty3}{\showpaste}
\tab{5}\spadcommand{1 + 2 - 3/ (4 * 3 ^ (2 - 1))}
\end{paste}\end{patch}

\begin{patch}{ugIntroConversionPagePatch1}
\begin{paste}{ugIntroConversionPageFull1}{ugIntroConversionPageEmpty1}
\pastebutton{ugIntroConversionPageFull1}{\hidepaste}
\tab{5}\spadcommand{p := r^2 + 2/3\bound{p }}
\indentrel{3}\begin{verbatim}
         2   2
   (1)  r  + -
             3
                    Type: Polynomial(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroConversionPageEmpty1}
\begin{paste}{ugIntroConversionPageEmpty1}{ugIntroConversionPagePatch1}
\pastebutton{ugIntroConversionPageEmpty1}{\showpaste}
\tab{5}\spadcommand{p := r^2 + 2/3\bound{p }}
\end{paste}\end{patch}

\begin{patch}{ugIntroConversionPagePatch2}
\begin{paste}{ugIntroConversionPageFull2}{ugIntroConversionPageEmpty2}
\pastebutton{ugIntroConversionPageFull2}{\hidepaste}
\tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }}
\indentrel{3}\begin{verbatim}
           2
        3 r  + 2
   (2)  --------
            3
                    Type: Fraction(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroConversionPageEmpty2}
\begin{paste}{ugIntroConversionPageEmpty2}{ugIntroConversionPagePatch2}
\pastebutton{ugIntroConversionPageEmpty2}{\showpaste}
\tab{5}\spadcommand{p :: Fraction Polynomial Integer\free{p }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch1}
\begin{paste}{ugIntroYouPageFull1}{ugIntroYouPageEmpty1}
\pastebutton{ugIntroYouPageFull1}{\hidepaste}
\tab{5}\spadcommand{fact(0) == 1\bound{fact }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty1}
\begin{paste}{ugIntroYouPageEmpty1}{ugIntroYouPagePatch1}
\pastebutton{ugIntroYouPageEmpty1}{\showpaste}
\tab{5}\spadcommand{fact(0) == 1\bound{fact }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch2}
\begin{paste}{ugIntroYouPageFull2}{ugIntroYouPageEmpty2}
\pastebutton{ugIntroYouPageFull2}{\hidepaste}
\tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty2}
\begin{paste}{ugIntroYouPageEmpty2}{ugIntroYouPagePatch2}
\pastebutton{ugIntroYouPageEmpty2}{\showpaste}
\tab{5}\spadcommand{fact(n) == n*fact(n-1)\bound{facta }\free{fact }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch3}
\begin{paste}{ugIntroYouPageFull3}{ugIntroYouPageEmpty3}
\pastebutton{ugIntroYouPageFull3}{\hidepaste}
\tab{5}\spadcommand{fact(50)\free{facta }}
\indentrel{3}\begin{verbatim}
   (3)
  304140932017133780436126081660647688443776415689605120_
   00000000000
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty3}
\begin{paste}{ugIntroYouPageEmpty3}{ugIntroYouPagePatch3}
\pastebutton{ugIntroYouPageEmpty3}{\showpaste}
\tab{5}\spadcommand{fact(50)\free{facta }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch4}
\begin{paste}{ugIntroYouPageFull4}{ugIntroYouPageEmpty4}
\pastebutton{ugIntroYouPageFull4}{\hidepaste}
\tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty4}
\begin{paste}{ugIntroYouPageEmpty4}{ugIntroYouPagePatch4}
\pastebutton{ugIntroYouPageEmpty4}{\showpaste}
\tab{5}\spadcommand{fac(n) == if n < 3 then n else n * fac(n - 1)\bound{fac }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch5}
\begin{paste}{ugIntroYouPageFull5}{ugIntroYouPageEmpty5}
\pastebutton{ugIntroYouPageFull5}{\hidepaste}
\tab{5}\spadcommand{fac(50)\free{fac }}
\indentrel{3}\begin{verbatim}
   (5)
  304140932017133780436126081660647688443776415689605120_
   00000000000
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty5}
\begin{paste}{ugIntroYouPageEmpty5}{ugIntroYouPagePatch5}
\pastebutton{ugIntroYouPageEmpty5}{\showpaste}
\tab{5}\spadcommand{fac(50)\free{fac }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch6}
\begin{paste}{ugIntroYouPageFull6}{ugIntroYouPageEmpty6}
\pastebutton{ugIntroYouPageFull6}{\hidepaste}
\tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty6}
\begin{paste}{ugIntroYouPageEmpty6}{ugIntroYouPagePatch6}
\pastebutton{ugIntroYouPageEmpty6}{\showpaste}
\tab{5}\spadcommand{fa(n) == (a := 1; for i in 2..n repeat a := a*i; a)\bound{fa }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch7}
\begin{paste}{ugIntroYouPageFull7}{ugIntroYouPageEmpty7}
\pastebutton{ugIntroYouPageFull7}{\hidepaste}
\tab{5}\spadcommand{fa(50)\free{fa }}
\indentrel{3}\begin{verbatim}
   (7)
  304140932017133780436126081660647688443776415689605120_
   00000000000
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty7}
\begin{paste}{ugIntroYouPageEmpty7}{ugIntroYouPagePatch7}
\pastebutton{ugIntroYouPageEmpty7}{\showpaste}
\tab{5}\spadcommand{fa(50)\free{fa }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch8}
\begin{paste}{ugIntroYouPageFull8}{ugIntroYouPageEmpty8}
\pastebutton{ugIntroYouPageFull8}{\hidepaste}
\tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty8}
\begin{paste}{ugIntroYouPageEmpty8}{ugIntroYouPagePatch8}
\pastebutton{ugIntroYouPageEmpty8}{\showpaste}
\tab{5}\spadcommand{f(n) == reduce(*,[i for i in 2..n])\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch9}
\begin{paste}{ugIntroYouPageFull9}{ugIntroYouPageEmpty9}
\pastebutton{ugIntroYouPageFull9}{\hidepaste}
\tab{5}\spadcommand{f(50)\free{f }}
\indentrel{3}\begin{verbatim}
   (9)
  304140932017133780436126081660647688443776415689605120_
   00000000000
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty9}
\begin{paste}{ugIntroYouPageEmpty9}{ugIntroYouPagePatch9}
\pastebutton{ugIntroYouPageEmpty9}{\showpaste}
\tab{5}\spadcommand{f(50)\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch10}
\begin{paste}{ugIntroYouPageFull10}{ugIntroYouPageEmpty10}
\pastebutton{ugIntroYouPageFull10}{\hidepaste}
\tab{5}\spadcommand{factorial(50)}
\indentrel{3}\begin{verbatim}
   (10)
  304140932017133780436126081660647688443776415689605120_
   00000000000
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty10}
\begin{paste}{ugIntroYouPageEmpty10}{ugIntroYouPagePatch10}
\pastebutton{ugIntroYouPageEmpty10}{\showpaste}
\tab{5}\spadcommand{factorial(50)}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch11}
\begin{paste}{ugIntroYouPageFull11}{ugIntroYouPageEmpty11}
\pastebutton{ugIntroYouPageFull11}{\hidepaste}
\tab{5}\spadcommand{permMat(n, i, j) ==
  m := diagonalMatrix
    [(if i = k or j = k then 0 else 1)
      for k in 1..n]
  m(i,j) := 1
  m(j,i) := 1
  m
\bound{permMat }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty11}
\begin{paste}{ugIntroYouPageEmpty11}{ugIntroYouPagePatch11}
\pastebutton{ugIntroYouPageEmpty11}{\showpaste}
\tab{5}\spadcommand{permMat(n, i, j) ==
  m := diagonalMatrix
    [(if i = k or j = k then 0 else 1)
      for k in 1..n]
  m(i,j) := 1
  m(j,i) := 1
  m
\bound{permMat }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch12}
\begin{paste}{ugIntroYouPageFull12}{ugIntroYouPageEmpty12}
\pastebutton{ugIntroYouPageFull12}{\hidepaste}
\tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }}
\indentrel{3}\begin{verbatim}
         +1  0  0  0+
         |          |
         |0  0  1  0|
   (12)  |          |
         |0  1  0  0|
         |          |
         +0  0  0  1+
                       Type: Matrix(NonNegativeInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty12}
\begin{paste}{ugIntroYouPageEmpty12}{ugIntroYouPagePatch12}
\pastebutton{ugIntroYouPageEmpty12}{\showpaste}
\tab{5}\spadcommand{p := permMat(4,2,3)\free{permMat }\bound{p }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch13}
\begin{paste}{ugIntroYouPageFull13}{ugIntroYouPageEmpty13}
\pastebutton{ugIntroYouPageFull13}{\hidepaste}
\tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }}
\indentrel{3}\begin{verbatim}
         +1   2   3   4 +
         |              |
         |5   6   7   8 |
   (13)  |              |
         |9   10  11  12|
         |              |
         +13  14  15  16+
                       Type: Matrix(NonNegativeInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty13}
\begin{paste}{ugIntroYouPageEmpty13}{ugIntroYouPagePatch13}
\pastebutton{ugIntroYouPageEmpty13}{\showpaste}
\tab{5}\spadcommand{m := matrix [[4*i + j for j in 1..4] for i in 0..3]\bound{m }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch14}
\begin{paste}{ugIntroYouPageFull14}{ugIntroYouPageEmpty14}
\pastebutton{ugIntroYouPageFull14}{\hidepaste}
\tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }}
\indentrel{3}\begin{verbatim}
         +1   2   3   4 +
         |              |
         |9   10  11  12|
   (14)  |              |
         |5   6   7   8 |
         |              |
         +13  14  15  16+
                       Type: Matrix(NonNegativeInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty14}
\begin{paste}{ugIntroYouPageEmpty14}{ugIntroYouPagePatch14}
\pastebutton{ugIntroYouPageEmpty14}{\showpaste}
\tab{5}\spadcommand{permMat(4,2,3) * m\free{p m }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch15}
\begin{paste}{ugIntroYouPageFull15}{ugIntroYouPageEmpty15}
\pastebutton{ugIntroYouPageFull15}{\hidepaste}
\tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty15}
\begin{paste}{ugIntroYouPageEmpty15}{ugIntroYouPagePatch15}
\pastebutton{ugIntroYouPageEmpty15}{\showpaste}
\tab{5}\spadcommand{t : (Float -> Float, Float) -> Float\bound{tdecl }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch16}
\begin{paste}{ugIntroYouPageFull16}{ugIntroYouPageEmpty16}
\pastebutton{ugIntroYouPageFull16}{\hidepaste}
\tab{5}\spadcommand{t(fun, x) == fun(x)^2 + sin(x)^2\free{tdecl }\bound{t }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty16}
\begin{paste}{ugIntroYouPageEmpty16}{ugIntroYouPagePatch16}
\pastebutton{ugIntroYouPageEmpty16}{\showpaste}
\tab{5}\spadcommand{t(fun, x) == fun(x)^2 + sin(x)^2\free{tdecl }\bound{t }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch17}
\begin{paste}{ugIntroYouPageFull17}{ugIntroYouPageEmpty17}
\pastebutton{ugIntroYouPageFull17}{\hidepaste}
\tab{5}\spadcommand{t(cos, 5.2058)\free{t }}
\indentrel{3}\begin{verbatim}
   (17)  1.0
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty17}
\begin{paste}{ugIntroYouPageEmpty17}{ugIntroYouPagePatch17}
\pastebutton{ugIntroYouPageEmpty17}{\showpaste}
\tab{5}\spadcommand{t(cos, 5.2058)\free{t }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch18}
\begin{paste}{ugIntroYouPageFull18}{ugIntroYouPageEmpty18}
\pastebutton{ugIntroYouPageFull18}{\hidepaste}
\tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty18}
\begin{paste}{ugIntroYouPageEmpty18}{ugIntroYouPagePatch18}
\pastebutton{ugIntroYouPageEmpty18}{\showpaste}
\tab{5}\spadcommand{cosinv(y) == cos(1/y)\bound{cosinv }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch19}
\begin{paste}{ugIntroYouPageFull19}{ugIntroYouPageEmpty19}
\pastebutton{ugIntroYouPageFull19}{\hidepaste}
\tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }}
\indentrel{3}\begin{verbatim}
   (19)  1.7392237241_800516493
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty19}
\begin{paste}{ugIntroYouPageEmpty19}{ugIntroYouPagePatch19}
\pastebutton{ugIntroYouPageEmpty19}{\showpaste}
\tab{5}\spadcommand{t(cosinv, 5.2058)\free{t }\free{cosinv }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch20}
\begin{paste}{ugIntroYouPageFull20}{ugIntroYouPageEmpty20}
\pastebutton{ugIntroYouPageFull20}{\hidepaste}
\tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }}
\indentrel{3}\begin{verbatim}
             +-+ +-+        +---+
   (20)  %DD\|a \|b  == %DD\|a b
 Type: RewriteRule(Integer,Integer,Expression(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty20}
\begin{paste}{ugIntroYouPageEmpty20}{ugIntroYouPagePatch20}
\pastebutton{ugIntroYouPageEmpty20}{\showpaste}
\tab{5}\spadcommand{groupSqrt := rule(sqrt(a) * sqrt(b) == sqrt(a*b))\bound{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch21}
\begin{paste}{ugIntroYouPageFull21}{ugIntroYouPageEmpty21}
\pastebutton{ugIntroYouPageFull21}{\hidepaste}
\tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))^4\bound{sxy }}
\indentrel{3}\begin{verbatim}
   (21)
                         +-+                      +-+  +-+
     ((4 z + 4 y + 12 x)\|y  + (4 z + 12 y + 4 x)\|x )\|z
   + 
                        +-+ +-+    2                   2
     (12 z + 4 y + 4 x)\|x \|y  + z  + (6 y + 6 x)z + y
   + 
              2
     6 x y + x
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty21}
\begin{paste}{ugIntroYouPageEmpty21}{ugIntroYouPagePatch21}
\pastebutton{ugIntroYouPageEmpty21}{\showpaste}
\tab{5}\spadcommand{a := (sqrt(x) + sqrt(y) + sqrt(z))^4\bound{sxy }}
\end{paste}\end{patch}

\begin{patch}{ugIntroYouPagePatch22}
\begin{paste}{ugIntroYouPageFull22}{ugIntroYouPageEmpty22}
\pastebutton{ugIntroYouPageFull22}{\hidepaste}
\tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }}
\indentrel{3}\begin{verbatim}
   (22)
                        +---+                      +---+
     (4 z + 4 y + 12 x)\|y z  + (4 z + 12 y + 4 x)\|x z
   + 
                        +---+    2                   2
     (12 z + 4 y + 4 x)\|x y  + z  + (6 y + 6 x)z + y
   + 
              2
     6 x y + x
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroYouPageEmpty22}
\begin{paste}{ugIntroYouPageEmpty22}{ugIntroYouPagePatch22}
\pastebutton{ugIntroYouPageEmpty22}{\showpaste}
\tab{5}\spadcommand{groupSqrt a\free{sxy }\free{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPagePatch1}
\begin{paste}{ugIntroTypesPageFull1}{ugIntroTypesPageEmpty1}
\pastebutton{ugIntroTypesPageFull1}{\hidepaste}
\tab{5}\spadcommand{8}
\indentrel{3}\begin{verbatim}
   (1)  8
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPageEmpty1}
\begin{paste}{ugIntroTypesPageEmpty1}{ugIntroTypesPagePatch1}
\pastebutton{ugIntroTypesPageEmpty1}{\showpaste}
\tab{5}\spadcommand{8}
\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPagePatch2}
\begin{paste}{ugIntroTypesPageFull2}{ugIntroTypesPageEmpty2}
\pastebutton{ugIntroTypesPageFull2}{\hidepaste}
\tab{5}\spadcommand{-8}
\indentrel{3}\begin{verbatim}
   (2)  - 8
                                          Type: Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPageEmpty2}
\begin{paste}{ugIntroTypesPageEmpty2}{ugIntroTypesPagePatch2}
\pastebutton{ugIntroTypesPageEmpty2}{\showpaste}
\tab{5}\spadcommand{-8}
\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPagePatch3}
\begin{paste}{ugIntroTypesPageFull3}{ugIntroTypesPageEmpty3}
\pastebutton{ugIntroTypesPageFull3}{\hidepaste}
\tab{5}\spadcommand{x^8}
\indentrel{3}\begin{verbatim}
         8
   (3)  x
                              Type: Polynomial(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPageEmpty3}
\begin{paste}{ugIntroTypesPageEmpty3}{ugIntroTypesPagePatch3}
\pastebutton{ugIntroTypesPageEmpty3}{\showpaste}
\tab{5}\spadcommand{x^8}
\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPagePatch4}
\begin{paste}{ugIntroTypesPageFull4}{ugIntroTypesPageEmpty4}
\pastebutton{ugIntroTypesPageFull4}{\hidepaste}
\tab{5}\spadcommand{x^(-8)}
\indentrel{3}\begin{verbatim}
         1
   (4)  --
         8
        x
                    Type: Fraction(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTypesPageEmpty4}
\begin{paste}{ugIntroTypesPageEmpty4}{ugIntroTypesPagePatch4}
\pastebutton{ugIntroTypesPageEmpty4}{\showpaste}
\tab{5}\spadcommand{x^(-8)}
\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPagePatch1}
\begin{paste}{ugIntroTwoDimPageFull1}{ugIntroTwoDimPageEmpty1}
\pastebutton{ugIntroTwoDimPageFull1}{\hidepaste}
\tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }}
\indentrel{3}\begin{verbatim}
        +1  2+
   (1)  |    |
        +3  4+
                                  Type: Matrix(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPageEmpty1}
\begin{paste}{ugIntroTwoDimPageEmpty1}{ugIntroTwoDimPagePatch1}
\pastebutton{ugIntroTwoDimPageEmpty1}{\showpaste}
\tab{5}\spadcommand{m := matrix([[1,2], [3,4]])\bound{m }}
\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPagePatch2}
\begin{paste}{ugIntroTwoDimPageFull2}{ugIntroTwoDimPageEmpty2}
\pastebutton{ugIntroTwoDimPageFull2}{\hidepaste}
\tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }}
\indentrel{3}\begin{verbatim}
        +    1        1        1        1  +
        |- -----  - -----  - -----  - -----|
        |  x - 2    x - 3    x - 4    x - 5|
        |                                  |
        |    1        1        1        1  |
        |- -----  - -----  - -----  - -----|
        |  x - 3    x - 4    x - 5    x - 6|
   (2)  |                                  |
        |    1        1        1        1  |
        |- -----  - -----  - -----  - -----|
        |  x - 4    x - 5    x - 6    x - 7|
        |                                  |
        |    1        1        1        1  |
        |- -----  - -----  - -----  - -----|
        +  x - 5    x - 6    x - 7    x - 8+
            Type: Matrix(Fraction(Polynomial(Integer)))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPageEmpty2}
\begin{paste}{ugIntroTwoDimPageEmpty2}{ugIntroTwoDimPagePatch2}
\pastebutton{ugIntroTwoDimPageEmpty2}{\showpaste}
\tab{5}\spadcommand{matrix([[1/(i + j - x) for i in 1..4] for j in 1..4])\bound{hilb }}
\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPagePatch3}
\begin{paste}{ugIntroTwoDimPageFull3}{ugIntroTwoDimPageEmpty3}
\pastebutton{ugIntroTwoDimPageFull3}{\hidepaste}
\tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }}
\indentrel{3}\begin{verbatim}
        +1   1   1 +
        |          |
   (3)  |x   y   z |
        |          |
        | 2   2   2|
        +x   y   z +
                      Type: Matrix(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPageEmpty3}
\begin{paste}{ugIntroTwoDimPageEmpty3}{ugIntroTwoDimPagePatch3}
\pastebutton{ugIntroTwoDimPageEmpty3}{\showpaste}
\tab{5}\spadcommand{vm := matrix [[1,1,1], [x,y,z], [x*x,y*y,z*z]]\bound{vm }}
\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPagePatch4}
\begin{paste}{ugIntroTwoDimPageFull4}{ugIntroTwoDimPageEmpty4}
\pastebutton{ugIntroTwoDimPageFull4}{\hidepaste}
\tab{5}\spadcommand{vm(3,3)\free{vm }}
\indentrel{3}\begin{verbatim}
         2
   (4)  z
                              Type: Polynomial(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPageEmpty4}
\begin{paste}{ugIntroTwoDimPageEmpty4}{ugIntroTwoDimPagePatch4}
\pastebutton{ugIntroTwoDimPageEmpty4}{\showpaste}
\tab{5}\spadcommand{vm(3,3)\free{vm }}
\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPagePatch5}
\begin{paste}{ugIntroTwoDimPageFull5}{ugIntroTwoDimPageEmpty5}
\pastebutton{ugIntroTwoDimPageFull5}{\hidepaste}
\tab{5}\spadcommand{column(vm,2)\free{vm }}
\indentrel{3}\begin{verbatim}
                2
   (5)  [1, y, y ]
                      Type: Vector(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPageEmpty5}
\begin{paste}{ugIntroTwoDimPageEmpty5}{ugIntroTwoDimPagePatch5}
\pastebutton{ugIntroTwoDimPageEmpty5}{\showpaste}
\tab{5}\spadcommand{column(vm,2)\free{vm }}
\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPagePatch6}
\begin{paste}{ugIntroTwoDimPageFull6}{ugIntroTwoDimPageEmpty6}
\pastebutton{ugIntroTwoDimPageFull6}{\hidepaste}
\tab{5}\spadcommand{vm * vm\free{vm }}
\indentrel{3}\begin{verbatim}
   (6)
   +    2                2              2          +
   |   x  + x + 1       y  + y + 1     z  + z + 1  |
   |                                               |
   |  2                 2     2        3           |
   | x z + x y + x     y z + y  + x   z  + y z + x |
   |                                               |
   | 2 2      2    2   2 2    3    2   4    2     2|
   +x z  + x y  + x   y z  + y  + x   z  + y z + x +
                      Type: Matrix(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPageEmpty6}
\begin{paste}{ugIntroTwoDimPageEmpty6}{ugIntroTwoDimPagePatch6}
\pastebutton{ugIntroTwoDimPageEmpty6}{\showpaste}
\tab{5}\spadcommand{vm * vm\free{vm }}
\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPagePatch7}
\begin{paste}{ugIntroTwoDimPageFull7}{ugIntroTwoDimPageEmpty7}
\pastebutton{ugIntroTwoDimPageFull7}{\hidepaste}
\tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }}
\indentrel{3}\begin{verbatim}
   (7)  (y - x)(z - y)(z - x)
                    Type: Factored(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroTwoDimPageEmpty7}
\begin{paste}{ugIntroTwoDimPageEmpty7}{ugIntroTwoDimPagePatch7}
\pastebutton{ugIntroTwoDimPageEmpty7}{\showpaste}
\tab{5}\spadcommand{factor determinant vm\free{vm }\bound{d }}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch1}
\begin{paste}{ugIntroAssignPageFull1}{ugIntroAssignPageEmpty1}
\pastebutton{ugIntroAssignPageFull1}{\hidepaste}
\tab{5}\spadcommand{(x - y*z)^2}
\indentrel{3}\begin{verbatim}
         2 2              2
   (1)  y z  - 2 x y z + x
                              Type: Polynomial(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty1}
\begin{paste}{ugIntroAssignPageEmpty1}{ugIntroAssignPagePatch1}
\pastebutton{ugIntroAssignPageEmpty1}{\showpaste}
\tab{5}\spadcommand{(x - y*z)^2}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch2}
\begin{paste}{ugIntroAssignPageFull2}{ugIntroAssignPageEmpty2}
\pastebutton{ugIntroAssignPageFull2}{\hidepaste}
\tab{5}\spadcommand{x := 4}
\indentrel{3}\begin{verbatim}
   (2)  4
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty2}
\begin{paste}{ugIntroAssignPageEmpty2}{ugIntroAssignPagePatch2}
\pastebutton{ugIntroAssignPageEmpty2}{\showpaste}
\tab{5}\spadcommand{x := 4}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch3}
\begin{paste}{ugIntroAssignPageFull3}{ugIntroAssignPageEmpty3}
\pastebutton{ugIntroAssignPageFull3}{\hidepaste}
\tab{5}\spadcommand{x := z + 3/5}
\indentrel{3}\begin{verbatim}
            3
   (3)  z + -
            5
                    Type: Polynomial(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty3}
\begin{paste}{ugIntroAssignPageEmpty3}{ugIntroAssignPagePatch3}
\pastebutton{ugIntroAssignPageEmpty3}{\showpaste}
\tab{5}\spadcommand{x := z + 3/5}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch4}
\begin{paste}{ugIntroAssignPageFull4}{ugIntroAssignPageEmpty4}
\pastebutton{ugIntroAssignPageFull4}{\hidepaste}
\tab{5}\spadcommand{y : Integer\bound{y }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty4}
\begin{paste}{ugIntroAssignPageEmpty4}{ugIntroAssignPagePatch4}
\pastebutton{ugIntroAssignPageEmpty4}{\showpaste}
\tab{5}\spadcommand{y : Integer\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch5}
\begin{paste}{ugIntroAssignPageFull5}{ugIntroAssignPageEmpty5}
\pastebutton{ugIntroAssignPageFull5}{\hidepaste}
\tab{5}\spadcommand{y := 89\bound{y1 }\free{y }}
\indentrel{3}\begin{verbatim}
   (5)  89
                                          Type: Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty5}
\begin{paste}{ugIntroAssignPageEmpty5}{ugIntroAssignPagePatch5}
\pastebutton{ugIntroAssignPageEmpty5}{\showpaste}
\tab{5}\spadcommand{y := 89\bound{y1 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch6}
\begin{paste}{ugIntroAssignPageFull6}{ugIntroAssignPageEmpty6}
\pastebutton{ugIntroAssignPageFull6}{\hidepaste}
\tab{5}\spadcommand{y := sin \%pi}
\indentrel{3}\begin{verbatim}
   (6)  0
                                          Type: Integer
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty6}
\begin{paste}{ugIntroAssignPageEmpty6}{ugIntroAssignPagePatch6}
\pastebutton{ugIntroAssignPageEmpty6}{\showpaste}
\tab{5}\spadcommand{y := sin \%pi}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch7}
\begin{paste}{ugIntroAssignPageFull7}{ugIntroAssignPageEmpty7}
\pastebutton{ugIntroAssignPageFull7}{\hidepaste}
\tab{5}\spadcommand{y := 2/3}
\indentrel{3}\begin{verbatim}
error
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty7}
\begin{paste}{ugIntroAssignPageEmpty7}{ugIntroAssignPagePatch7}
\pastebutton{ugIntroAssignPageEmpty7}{\showpaste}
\tab{5}\spadcommand{y := 2/3}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch8}
\begin{paste}{ugIntroAssignPageFull8}{ugIntroAssignPageEmpty8}
\pastebutton{ugIntroAssignPageFull8}{\hidepaste}
\tab{5}\spadcommand{f : Float := 2/3}
\indentrel{3}\begin{verbatim}
   (7)  0.6666666666_6666666667
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty8}
\begin{paste}{ugIntroAssignPageEmpty8}{ugIntroAssignPagePatch8}
\pastebutton{ugIntroAssignPageEmpty8}{\showpaste}
\tab{5}\spadcommand{f : Float := 2/3}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch9}
\begin{paste}{ugIntroAssignPageFull9}{ugIntroAssignPageEmpty9}
\pastebutton{ugIntroAssignPageFull9}{\hidepaste}
\tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }}
\indentrel{3}\begin{verbatim}
   (8)  0.6666666666_6666666667
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty9}
\begin{paste}{ugIntroAssignPageEmpty9}{ugIntroAssignPagePatch9}
\pastebutton{ugIntroAssignPageEmpty9}{\showpaste}
\tab{5}\spadcommand{f : Float; f := 2/3\bound{fff }}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch10}
\begin{paste}{ugIntroAssignPageFull10}{ugIntroAssignPageEmpty10}
\pastebutton{ugIntroAssignPageFull10}{\hidepaste}
\tab{5}\spadcommand{q}
\indentrel{3}\begin{verbatim}
   (9)  q
                                      Type: Variable(q)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty10}
\begin{paste}{ugIntroAssignPageEmpty10}{ugIntroAssignPagePatch10}
\pastebutton{ugIntroAssignPageEmpty10}{\showpaste}
\tab{5}\spadcommand{q}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch11}
\begin{paste}{ugIntroAssignPageFull11}{ugIntroAssignPageEmpty11}
\pastebutton{ugIntroAssignPageFull11}{\hidepaste}
\tab{5}\spadcommand{[q, r]}
\indentrel{3}\begin{verbatim}
   (10)  [q, r]
                 Type: List(OrderedVariableList([q,r]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty11}
\begin{paste}{ugIntroAssignPageEmpty11}{ugIntroAssignPagePatch11}
\pastebutton{ugIntroAssignPageEmpty11}{\showpaste}
\tab{5}\spadcommand{[q, r]}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch12}
\begin{paste}{ugIntroAssignPageFull12}{ugIntroAssignPageEmpty12}
\pastebutton{ugIntroAssignPageFull12}{\hidepaste}
\tab{5}\spadcommand{f\free{fff }}
\indentrel{3}\begin{verbatim}
   (11)  0.6666666666_6666666667
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty12}
\begin{paste}{ugIntroAssignPageEmpty12}{ugIntroAssignPagePatch12}
\pastebutton{ugIntroAssignPageEmpty12}{\showpaste}
\tab{5}\spadcommand{f\free{fff }}
\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPagePatch13}
\begin{paste}{ugIntroAssignPageFull13}{ugIntroAssignPageEmpty13}
\pastebutton{ugIntroAssignPageFull13}{\hidepaste}
\tab{5}\spadcommand{'f}
\indentrel{3}\begin{verbatim}
   (12)  f
                                      Type: Variable(f)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroAssignPageEmpty13}
\begin{paste}{ugIntroAssignPageEmpty13}{ugIntroAssignPagePatch13}
\pastebutton{ugIntroAssignPageEmpty13}{\showpaste}
\tab{5}\spadcommand{'f}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch1}
\begin{paste}{ugIntroSeriesPageFull1}{ugIntroSeriesPageEmpty1}
\pastebutton{ugIntroSeriesPageFull1}{\hidepaste}
\tab{5}\spadcommand{series(sin(a*x),x = 0)}
\indentrel{3}\begin{verbatim}
   (1)
            3        5        7          9
           a   3    a   5    a    7     a     9
     a x - -- x  + --- x  - ---- x  + ------ x
            6      120      5040      362880
   + 
           11
          a      11      12
     - -------- x   + O(x  )
       39916800
 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty1}
\begin{paste}{ugIntroSeriesPageEmpty1}{ugIntroSeriesPagePatch1}
\pastebutton{ugIntroSeriesPageEmpty1}{\showpaste}
\tab{5}\spadcommand{series(sin(a*x),x = 0)}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch2}
\begin{paste}{ugIntroSeriesPageFull2}{ugIntroSeriesPageEmpty2}
\pastebutton{ugIntroSeriesPageFull2}{\hidepaste}
\tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)}
\indentrel{3}\begin{verbatim}
   (2)
         a %pi          a %pi      %pi
     sin(-----) + a cos(-----)(x - ---)
           4              4         4
   + 
        2    a %pi                3    a %pi
       a sin(-----)              a cos(-----)
               4         %pi 2           4         %pi 3
     - ------------ (x - ---)  - ------------ (x - ---)
             2            4            6            4
   + 
      4    a %pi                5    a %pi
     a sin(-----)              a cos(-----)
             4         %pi 4           4         %pi 5
     ------------ (x - ---)  + ------------ (x - ---)
          24            4           120           4
   + 
        6    a %pi                7    a %pi
       a sin(-----)              a cos(-----)
               4         %pi 6           4         %pi 7
     - ------------ (x - ---)  - ------------ (x - ---)
            720           4          5040           4
   + 
      8    a %pi                9    a %pi
     a sin(-----)              a cos(-----)
             4         %pi 8           4         %pi 9
     ------------ (x - ---)  + ------------ (x - ---)
         40320          4         362880          4
   + 
        10    a %pi
       a  sin(-----)
                4         %pi 10          %pi 11
     - ------------- (x - ---)   + O((x - ---)  )
          3628800          4               4
Type: UnivariatePuiseuxSeries(Expression(Integer),x,%pi/4)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty2}
\begin{paste}{ugIntroSeriesPageEmpty2}{ugIntroSeriesPagePatch2}
\pastebutton{ugIntroSeriesPageEmpty2}{\showpaste}
\tab{5}\spadcommand{series(sin(a*x),x = \%pi/4)}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch3}
\begin{paste}{ugIntroSeriesPageFull3}{ugIntroSeriesPageEmpty3}
\pastebutton{ugIntroSeriesPageFull3}{\hidepaste}
\tab{5}\spadcommand{series(n +-> (-1)^((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)}
\indentrel{3}\begin{verbatim}
         4      10
         -      --
         3   1   3      5
   (3)  x  - - x   + O(x )
             6
 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty3}
\begin{paste}{ugIntroSeriesPageEmpty3}{ugIntroSeriesPagePatch3}
\pastebutton{ugIntroSeriesPageEmpty3}{\showpaste}
\tab{5}\spadcommand{series(n +-> (-1)^((3*n - 4)/6)/factorial(n - 1/3),x = 0,4/3..,2)}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch4}
\begin{paste}{ugIntroSeriesPageFull4}{ugIntroSeriesPageEmpty4}
\pastebutton{ugIntroSeriesPageFull4}{\hidepaste}
\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
\indentrel{3}\begin{verbatim}
   (4)
              2    3    4    5    6    7    8    9    10
     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
   + 
        11
     O(x  )
 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty4}
\begin{paste}{ugIntroSeriesPageEmpty4}{ugIntroSeriesPagePatch4}
\pastebutton{ugIntroSeriesPageEmpty4}{\showpaste}
\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch5}
\begin{paste}{ugIntroSeriesPageFull5}{ugIntroSeriesPageEmpty5}
\pastebutton{ugIntroSeriesPageFull5}{\hidepaste}
\tab{5}\spadcommand{f ^ 2\free{f }}
\indentrel{3}\begin{verbatim}
   (5)
                  2      3      4      5      6      7
     1 + 2 x + 3 x  + 4 x  + 5 x  + 6 x  + 7 x  + 8 x
   + 
        8       9       10      11
     9 x  + 10 x  + 11 x   + O(x  )
 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty5}
\begin{paste}{ugIntroSeriesPageEmpty5}{ugIntroSeriesPagePatch5}
\pastebutton{ugIntroSeriesPageEmpty5}{\showpaste}
\tab{5}\spadcommand{f ^ 2\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch6}
\begin{paste}{ugIntroSeriesPageFull6}{ugIntroSeriesPageEmpty6}
\pastebutton{ugIntroSeriesPageFull6}{\hidepaste}
\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
\indentrel{3}\begin{verbatim}
   (6)
              2    3    4    5    6    7    8    9    10
     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
   + 
        11
     O(x  )
 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty6}
\begin{paste}{ugIntroSeriesPageEmpty6}{ugIntroSeriesPagePatch6}
\pastebutton{ugIntroSeriesPageEmpty6}{\showpaste}
\tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch7}
\begin{paste}{ugIntroSeriesPageFull7}{ugIntroSeriesPageEmpty7}
\pastebutton{ugIntroSeriesPageFull7}{\hidepaste}
\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
\indentrel{3}\begin{verbatim}
   (7)
         1  2   1  3   1  4   1  5   1  6   1  7   1  8
     x + - x  + - x  + - x  + - x  + - x  + - x  + - x
         2      3      4      5      6      7      8
   + 
     1  9    1  10    1  11      12
     - x  + -- x   + -- x   + O(x  )
     9      10       11
 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty7}
\begin{paste}{ugIntroSeriesPageEmpty7}{ugIntroSeriesPagePatch7}
\pastebutton{ugIntroSeriesPageEmpty7}{\showpaste}
\tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch8}
\begin{paste}{ugIntroSeriesPageFull8}{ugIntroSeriesPageEmpty8}
\pastebutton{ugIntroSeriesPageFull8}{\hidepaste}
\tab{5}\spadcommand{exp(g)\free{g }}
\indentrel{3}\begin{verbatim}
   (8)
              2    3    4    5    6    7    8    9    10
     1 + x + x  + x  + x  + x  + x  + x  + x  + x  + x
   + 
        11
     O(x  )
 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty8}
\begin{paste}{ugIntroSeriesPageEmpty8}{ugIntroSeriesPagePatch8}
\pastebutton{ugIntroSeriesPageEmpty8}{\showpaste}
\tab{5}\spadcommand{exp(g)\free{g }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch9}
\begin{paste}{ugIntroSeriesPageFull9}{ugIntroSeriesPageEmpty9}
\pastebutton{ugIntroSeriesPageFull9}{\hidepaste}
\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
\indentrel{3}\begin{verbatim}
   (9)
             1  2   1  3    1  4    1   5    1   6
     1 + x + - x  + - x  + -- x  + --- x  + --- x
             2      6      24      120      720
   + 
     1   7     1    8      1    9      1     10      11
   ---- x  + ----- x  + ------ x  + ------- x   + O(x  )
   5040      40320      362880      3628800
  Type: UnivariateTaylorSeries(Expression(Integer),x,0)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty9}
\begin{paste}{ugIntroSeriesPageEmpty9}{ugIntroSeriesPagePatch9}
\pastebutton{ugIntroSeriesPageEmpty9}{\showpaste}
\tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPagePatch10}
\begin{paste}{ugIntroSeriesPageFull10}{ugIntroSeriesPageEmpty10}
\pastebutton{ugIntroSeriesPageFull10}{\hidepaste}
\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
\indentrel{3}\begin{verbatim}
   (10)
   [1.0, 2.0, 2.5, 2.6666666666_666666667,
    2.7083333333_333333333, 2.7166666666_666666667,
    2.7180555555_555555556, 2.7182539682_53968254,
    2.7182787698_412698413, 2.7182815255_731922399, ...]
                        Type: Stream(Expression(Float))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSeriesPageEmpty10}
\begin{paste}{ugIntroSeriesPageEmpty10}{ugIntroSeriesPagePatch10}
\pastebutton{ugIntroSeriesPageEmpty10}{\showpaste}
\tab{5}\spadcommand{eval(f,1.0)\free{f2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPagePatch1}
\begin{paste}{ugIntroCallFunPageFull1}{ugIntroCallFunPageEmpty1}
\pastebutton{ugIntroCallFunPageFull1}{\hidepaste}
\tab{5}\spadcommand{factor(120)}
\indentrel{3}\begin{verbatim}
         3
   (1)  2 3 5
                                Type: Factored(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPageEmpty1}
\begin{paste}{ugIntroCallFunPageEmpty1}{ugIntroCallFunPagePatch1}
\pastebutton{ugIntroCallFunPageEmpty1}{\showpaste}
\tab{5}\spadcommand{factor(120)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPagePatch2}
\begin{paste}{ugIntroCallFunPageFull2}{ugIntroCallFunPageEmpty2}
\pastebutton{ugIntroCallFunPageFull2}{\hidepaste}
\tab{5}\spadcommand{divide(125,7)}
\indentrel{3}\begin{verbatim}
   (2)  [quotient = 17, remainder = 6]
     Type: Record(quotient: Integer,remainder: Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPageEmpty2}
\begin{paste}{ugIntroCallFunPageEmpty2}{ugIntroCallFunPagePatch2}
\pastebutton{ugIntroCallFunPageEmpty2}{\showpaste}
\tab{5}\spadcommand{divide(125,7)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPagePatch3}
\begin{paste}{ugIntroCallFunPageFull3}{ugIntroCallFunPageEmpty3}
\pastebutton{ugIntroCallFunPageFull3}{\hidepaste}
\tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)}
\indentrel{3}\begin{verbatim}
   (3)  3.4 + 5.6 i + 2.9 j + 0.1 k
                                Type: Quaternion(Float)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPageEmpty3}
\begin{paste}{ugIntroCallFunPageEmpty3}{ugIntroCallFunPagePatch3}
\pastebutton{ugIntroCallFunPageEmpty3}{\showpaste}
\tab{5}\spadcommand{quatern(3.4,5.6,2.9,0.1)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPagePatch4}
\begin{paste}{ugIntroCallFunPageFull4}{ugIntroCallFunPageEmpty4}
\pastebutton{ugIntroCallFunPageFull4}{\hidepaste}
\tab{5}\spadcommand{factorial 10}
\indentrel{3}\begin{verbatim}
   (4)  3628800
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCallFunPageEmpty4}
\begin{paste}{ugIntroCallFunPageEmpty4}{ugIntroCallFunPagePatch4}
\pastebutton{ugIntroCallFunPageEmpty4}{\showpaste}
\tab{5}\spadcommand{factorial 10}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch1}
\begin{paste}{ugIntroCollectPageFull1}{ugIntroCollectPageEmpty1}
\pastebutton{ugIntroCollectPageFull1}{\hidepaste}
\tab{5}\spadcommand{u := [1,-7,11]\bound{u }}
\indentrel{3}\begin{verbatim}
   (1)  [1, - 7, 11]
                                    Type: List(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty1}
\begin{paste}{ugIntroCollectPageEmpty1}{ugIntroCollectPagePatch1}
\pastebutton{ugIntroCollectPageEmpty1}{\showpaste}
\tab{5}\spadcommand{u := [1,-7,11]\bound{u }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch2}
\begin{paste}{ugIntroCollectPageFull2}{ugIntroCollectPageEmpty2}
\pastebutton{ugIntroCollectPageFull2}{\hidepaste}
\tab{5}\spadcommand{first rest rest u\free{u }}
\indentrel{3}\begin{verbatim}
   (2)  11
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty2}
\begin{paste}{ugIntroCollectPageEmpty2}{ugIntroCollectPagePatch2}
\pastebutton{ugIntroCollectPageEmpty2}{\showpaste}
\tab{5}\spadcommand{first rest rest u\free{u }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch3}
\begin{paste}{ugIntroCollectPageFull3}{ugIntroCollectPageEmpty3}
\pastebutton{ugIntroCollectPageFull3}{\hidepaste}
\tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }}
\indentrel{3}\begin{verbatim}
   (3)  [1, - 7, 11, 9, 1, 3, - 4]
                                    Type: List(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty3}
\begin{paste}{ugIntroCollectPageEmpty3}{ugIntroCollectPagePatch3}
\pastebutton{ugIntroCollectPageEmpty3}{\showpaste}
\tab{5}\spadcommand{concat!(u,[9,1,3,-4]); u\free{u }\bound{u1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch4}
\begin{paste}{ugIntroCollectPageFull4}{ugIntroCollectPageEmpty4}
\pastebutton{ugIntroCollectPageFull4}{\hidepaste}
\tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }}
\indentrel{3}\begin{verbatim}
   (4)  [9, 1, 3, - 4]
                                    Type: List(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty4}
\begin{paste}{ugIntroCollectPageEmpty4}{ugIntroCollectPagePatch4}
\pastebutton{ugIntroCollectPageEmpty4}{\showpaste}
\tab{5}\spadcommand{lastnode := rest(u,3)\free{u1 }\bound{u2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch5}
\begin{paste}{ugIntroCollectPageFull5}{ugIntroCollectPageEmpty5}
\pastebutton{ugIntroCollectPageFull5}{\hidepaste}
\tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }}
\indentrel{3}\begin{verbatim}
                 _____
   (5)  [1, - 7, 11, 9]
                                    Type: List(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty5}
\begin{paste}{ugIntroCollectPageEmpty5}{ugIntroCollectPagePatch5}
\pastebutton{ugIntroCollectPageEmpty5}{\showpaste}
\tab{5}\spadcommand{setrest!(lastnode,rest(u,2)); u\free{u2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch6}
\begin{paste}{ugIntroCollectPageFull6}{ugIntroCollectPageEmpty6}
\pastebutton{ugIntroCollectPageFull6}{\hidepaste}
\tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }}
\indentrel{3}\begin{verbatim}
   (6)
        2        3        2         4     2   2
   [2, 2 , 2 3, 2 , 2 5, 2 3, 2 7, 2 , 2 3 , 2 5, ...]
                        Type: Stream(Factored(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty6}
\begin{paste}{ugIntroCollectPageEmpty6}{ugIntroCollectPagePatch6}
\pastebutton{ugIntroCollectPageEmpty6}{\showpaste}
\tab{5}\spadcommand{[factor(i) for i in 2.. by 2]\bound{stream1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch7}
\begin{paste}{ugIntroCollectPageFull7}{ugIntroCollectPageEmpty7}
\pastebutton{ugIntroCollectPageFull7}{\hidepaste}
\tab{5}\spadcommand{\%.36\free{stream1 }}
\indentrel{3}\begin{verbatim}
         3 2
   (7)  2 3
                                Type: Factored(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty7}
\begin{paste}{ugIntroCollectPageEmpty7}{ugIntroCollectPagePatch7}
\pastebutton{ugIntroCollectPageEmpty7}{\showpaste}
\tab{5}\spadcommand{\%.36\free{stream1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch8}
\begin{paste}{ugIntroCollectPageFull8}{ugIntroCollectPageEmpty8}
\pastebutton{ugIntroCollectPageFull8}{\hidepaste}
\tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }}
\indentrel{3}\begin{verbatim}
                    3
   (8)  [1, - 7, 3, -]
                    2
           Type: OneDimensionalArray(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty8}
\begin{paste}{ugIntroCollectPageEmpty8}{ugIntroCollectPagePatch8}
\pastebutton{ugIntroCollectPageEmpty8}{\showpaste}
\tab{5}\spadcommand{a := oneDimensionalArray [1, -7, 3, 3/2]\bound{a }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch9}
\begin{paste}{ugIntroCollectPageFull9}{ugIntroCollectPageEmpty9}
\pastebutton{ugIntroCollectPageFull9}{\hidepaste}
\tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }}
\indentrel{3}\begin{verbatim}
                     3
   (9)  [1, - 7, 11, -]
                     2
           Type: OneDimensionalArray(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty9}
\begin{paste}{ugIntroCollectPageEmpty9}{ugIntroCollectPagePatch9}
\pastebutton{ugIntroCollectPageEmpty9}{\showpaste}
\tab{5}\spadcommand{a.3 := 11; a\bound{a1 }\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch10}
\begin{paste}{ugIntroCollectPageFull10}{ugIntroCollectPageEmpty10}
\pastebutton{ugIntroCollectPageFull10}{\hidepaste}
\tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }}
\indentrel{3}\begin{verbatim}
error
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty10}
\begin{paste}{ugIntroCollectPageEmpty10}{ugIntroCollectPagePatch10}
\pastebutton{ugIntroCollectPageEmpty10}{\showpaste}
\tab{5}\spadcommand{concat!(a,oneDimensionalArray [1,-2])\free{a1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch11}
\begin{paste}{ugIntroCollectPageFull11}{ugIntroCollectPageEmpty11}
\pastebutton{ugIntroCollectPageFull11}{\hidepaste}
\tab{5}\spadcommand{bits(32,true)}
\indentrel{3}\begin{verbatim}
   (10)  "11111111111111111111111111111111"
                                             Type: Bits
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty11}
\begin{paste}{ugIntroCollectPageEmpty11}{ugIntroCollectPagePatch11}
\pastebutton{ugIntroCollectPageEmpty11}{\showpaste}
\tab{5}\spadcommand{bits(32,true)}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch12}
\begin{paste}{ugIntroCollectPageFull12}{ugIntroCollectPageEmpty12}
\pastebutton{ugIntroCollectPageFull12}{\hidepaste}
\tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }}
\indentrel{3}\begin{verbatim}
   (11)  [2, 7, - 5]
                           Type: FlexibleArray(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty12}
\begin{paste}{ugIntroCollectPageEmpty12}{ugIntroCollectPagePatch12}
\pastebutton{ugIntroCollectPageEmpty12}{\showpaste}
\tab{5}\spadcommand{f := flexibleArray [2, 7, -5]\bound{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch13}
\begin{paste}{ugIntroCollectPageFull13}{ugIntroCollectPageEmpty13}
\pastebutton{ugIntroCollectPageFull13}{\hidepaste}
\tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,3)\free{f }}
\indentrel{3}\begin{verbatim}
   (12)  [2, 7, 11, - 3, - 5]
                           Type: FlexibleArray(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty13}
\begin{paste}{ugIntroCollectPageEmpty13}{ugIntroCollectPagePatch13}
\pastebutton{ugIntroCollectPageEmpty13}{\showpaste}
\tab{5}\spadcommand{insert!(flexibleArray [11, -3],f,3)\free{f }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch14}
\begin{paste}{ugIntroCollectPageFull14}{ugIntroCollectPageEmpty14}
\pastebutton{ugIntroCollectPageFull14}{\hidepaste}
\tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }}
\indentrel{3}\begin{verbatim}
   (13)  [11, 7, - 4, 3, 4, - 7]
                                    Type: Heap(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty14}
\begin{paste}{ugIntroCollectPageEmpty14}{ugIntroCollectPagePatch14}
\pastebutton{ugIntroCollectPageEmpty14}{\showpaste}
\tab{5}\spadcommand{h := heap [-4,7,11,3,4,-7]\bound{h }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch15}
\begin{paste}{ugIntroCollectPageFull15}{ugIntroCollectPageEmpty15}
\pastebutton{ugIntroCollectPageFull15}{\hidepaste}
\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }}
\indentrel{3}\begin{verbatim}
   (14)  [11, 7, 4, 3, - 4, - 7]
                                    Type: List(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty15}
\begin{paste}{ugIntroCollectPageEmpty15}{ugIntroCollectPagePatch15}
\pastebutton{ugIntroCollectPageEmpty15}{\showpaste}
\tab{5}\spadcommand{[extract!(h) while not empty?(h)]\free{h }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch16}
\begin{paste}{ugIntroCollectPageFull16}{ugIntroCollectPageEmpty16}
\pastebutton{ugIntroCollectPageFull16}{\hidepaste}
\tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]}
\indentrel{3}\begin{verbatim}
   (15)  [[2, 3, 4], 5, [7, 9, 11]]
                Type: BinarySearchTree(PositiveInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty16}
\begin{paste}{ugIntroCollectPageEmpty16}{ugIntroCollectPagePatch16}
\pastebutton{ugIntroCollectPageEmpty16}{\showpaste}
\tab{5}\spadcommand{binarySearchTree [5,3,2,9,4,7,11]}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch17}
\begin{paste}{ugIntroCollectPageFull17}{ugIntroCollectPageEmpty17}
\pastebutton{ugIntroCollectPageFull17}{\hidepaste}
\tab{5}\spadcommand{modTree(8,[2,3,5,7])}
\indentrel{3}\begin{verbatim}
   (16)  [0, 2, 3, 1]
                                    Type: List(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty17}
\begin{paste}{ugIntroCollectPageEmpty17}{ugIntroCollectPagePatch17}
\pastebutton{ugIntroCollectPageEmpty17}{\showpaste}
\tab{5}\spadcommand{modTree(8,[2,3,5,7])}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch18}
\begin{paste}{ugIntroCollectPageFull18}{ugIntroCollectPageEmpty18}
\pastebutton{ugIntroCollectPageFull18}{\hidepaste}
\tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }}
\indentrel{3}\begin{verbatim}
            1  1  4
   (17)  {- -, -, -}
            3  3  5
                           Type: Set(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty18}
\begin{paste}{ugIntroCollectPageEmpty18}{ugIntroCollectPagePatch18}
\pastebutton{ugIntroCollectPageEmpty18}{\showpaste}
\tab{5}\spadcommand{fs := set[1/3,4/5,-1/3,4/5]\bound{fs }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch19}
\begin{paste}{ugIntroCollectPageFull19}{ugIntroCollectPageEmpty19}
\pastebutton{ugIntroCollectPageFull19}{\hidepaste}
\tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]}
\indentrel{3}\begin{verbatim}
   (18)  {47 : 2, 42 : 3, 0, 40 : 1, 38 : 4}
                                Type: Multiset(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty19}
\begin{paste}{ugIntroCollectPageEmpty19}{ugIntroCollectPagePatch19}
\pastebutton{ugIntroCollectPageEmpty19}{\showpaste}
\tab{5}\spadcommand{multiset [x rem 5 for x in primes(2,1000)]}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch20}
\begin{paste}{ugIntroCollectPageFull20}{ugIntroCollectPageEmpty20}
\pastebutton{ugIntroCollectPageFull20}{\hidepaste}
\tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }}
\indentrel{3}\begin{verbatim}
   (19)  table()
                           Type: Table(Integer,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty20}
\begin{paste}{ugIntroCollectPageEmpty20}{ugIntroCollectPagePatch20}
\pastebutton{ugIntroCollectPageEmpty20}{\showpaste}
\tab{5}\spadcommand{t : Table(Integer,Integer) := empty()\bound{t }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch21}
\begin{paste}{ugIntroCollectPageFull21}{ugIntroCollectPageEmpty21}
\pastebutton{ugIntroCollectPageFull21}{\hidepaste}
\tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty21}
\begin{paste}{ugIntroCollectPageEmpty21}{ugIntroCollectPagePatch21}
\pastebutton{ugIntroCollectPageEmpty21}{\showpaste}
\tab{5}\spadcommand{howMany(k) == (n:=search(k,t); n case "failed" => 1; n+1)\bound{how }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch22}
\begin{paste}{ugIntroCollectPageFull22}{ugIntroCollectPageEmpty22}
\pastebutton{ugIntroCollectPageFull22}{\hidepaste}
\tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }}
\indentrel{3}\begin{verbatim}
   (21)  table(4 = 38,1 = 40,0 = 1,3 = 42,2 = 47)
                           Type: Table(Integer,Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty22}
\begin{paste}{ugIntroCollectPageEmpty22}{ugIntroCollectPagePatch22}
\pastebutton{ugIntroCollectPageEmpty22}{\showpaste}
\tab{5}\spadcommand{for p in primes(2,1000) repeat (m:= p rem 5; t.m:= howMany(m)); t\free{how t }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch23}
\begin{paste}{ugIntroCollectPageFull23}{ugIntroCollectPageEmpty23}
\pastebutton{ugIntroCollectPageFull23}{\hidepaste}
\tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty23}
\begin{paste}{ugIntroCollectPageEmpty23}{ugIntroCollectPagePatch23}
\pastebutton{ugIntroCollectPageEmpty23}{\showpaste}
\tab{5}\spadcommand{daniel : Record(age : Integer, salary : Float)\bound{danieldec }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch24}
\begin{paste}{ugIntroCollectPageFull24}{ugIntroCollectPageEmpty24}
\pastebutton{ugIntroCollectPageFull24}{\hidepaste}
\tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }}
\indentrel{3}\begin{verbatim}
   (23)  [age = 28, salary = 32005.12]
               Type: Record(age: Integer,salary: Float)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty24}
\begin{paste}{ugIntroCollectPageEmpty24}{ugIntroCollectPagePatch24}
\pastebutton{ugIntroCollectPageEmpty24}{\showpaste}
\tab{5}\spadcommand{daniel := [28, 32005.12]\free{danieldec }\bound{daniel }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch25}
\begin{paste}{ugIntroCollectPageFull25}{ugIntroCollectPageEmpty25}
\pastebutton{ugIntroCollectPageFull25}{\hidepaste}
\tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }}
\indentrel{3}\begin{verbatim}
   (24)  [age = 28, salary = 35000.0]
               Type: Record(age: Integer,salary: Float)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty25}
\begin{paste}{ugIntroCollectPageEmpty25}{ugIntroCollectPagePatch25}
\pastebutton{ugIntroCollectPageEmpty25}{\showpaste}
\tab{5}\spadcommand{daniel.salary := 35000; daniel\free{daniel }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch26}
\begin{paste}{ugIntroCollectPageFull26}{ugIntroCollectPageEmpty26}
\pastebutton{ugIntroCollectPageFull26}{\hidepaste}
\tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty26}
\begin{paste}{ugIntroCollectPageEmpty26}{ugIntroCollectPagePatch26}
\pastebutton{ugIntroCollectPageEmpty26}{\showpaste}
\tab{5}\spadcommand{dog: Union(licenseNumber: Integer, name: String)\bound{xint }}
\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPagePatch27}
\begin{paste}{ugIntroCollectPageFull27}{ugIntroCollectPageEmpty27}
\pastebutton{ugIntroCollectPageFull27}{\hidepaste}
\tab{5}\spadcommand{dog := "Whisper"\free{xint }}
\indentrel{3}\begin{verbatim}
   (26)  "Whisper"
                          Type: Union(name: String,...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCollectPageEmpty27}
\begin{paste}{ugIntroCollectPageEmpty27}{ugIntroCollectPagePatch27}
\pastebutton{ugIntroCollectPageEmpty27}{\showpaste}
\tab{5}\spadcommand{dog := "Whisper"\free{xint }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch1}
\begin{paste}{ugIntroNumbersPageFull1}{ugIntroNumbersPageEmpty1}
\pastebutton{ugIntroNumbersPageFull1}{\hidepaste}
\tab{5}\spadcommand{11^13 * 13^11 * 17^7 - 19^5 * 23^3}
\indentrel{3}\begin{verbatim}
   (1)  25387751112538918594666224484237298
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty1}
\begin{paste}{ugIntroNumbersPageEmpty1}{ugIntroNumbersPagePatch1}
\pastebutton{ugIntroNumbersPageEmpty1}{\showpaste}
\tab{5}\spadcommand{11^13 * 13^11 * 17^7 - 19^5 * 23^3}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch2}
\begin{paste}{ugIntroNumbersPageFull2}{ugIntroNumbersPageEmpty2}
\pastebutton{ugIntroNumbersPageFull2}{\hidepaste}
\tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }}
\indentrel{3}\begin{verbatim}
          13  11  7  5  3  2
   (2)  11  13  17 19 23 29
                                Type: Factored(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty2}
\begin{paste}{ugIntroNumbersPageEmpty2}{ugIntroNumbersPagePatch2}
\pastebutton{ugIntroNumbersPageEmpty2}{\showpaste}
\tab{5}\spadcommand{factor 643238070748569023720594412551704344145570763243\bound{ex1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch3}
\begin{paste}{ugIntroNumbersPageFull3}{ugIntroNumbersPageEmpty3}
\pastebutton{ugIntroNumbersPageFull3}{\hidepaste}
\tab{5}\spadcommand{\% * 12\free{ex1 }}
\indentrel{3}\begin{verbatim}
         2    13  11  7  5  3  2
   (3)  2 3 11  13  17 19 23 29
                                Type: Factored(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty3}
\begin{paste}{ugIntroNumbersPageEmpty3}{ugIntroNumbersPagePatch3}
\pastebutton{ugIntroNumbersPageEmpty3}{\showpaste}
\tab{5}\spadcommand{\% * 12\free{ex1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch4}
\begin{paste}{ugIntroNumbersPageFull4}{ugIntroNumbersPageEmpty4}
\pastebutton{ugIntroNumbersPageFull4}{\hidepaste}
\tab{5}\spadcommand{radix(25937424601,11)}
\indentrel{3}\begin{verbatim}
   (4)  10000000000
                               Type: RadixExpansion(11)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty4}
\begin{paste}{ugIntroNumbersPageEmpty4}{ugIntroNumbersPagePatch4}
\pastebutton{ugIntroNumbersPageEmpty4}{\showpaste}
\tab{5}\spadcommand{radix(25937424601,11)}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch5}
\begin{paste}{ugIntroNumbersPageFull5}{ugIntroNumbersPageEmpty5}
\pastebutton{ugIntroNumbersPageFull5}{\hidepaste}
\tab{5}\spadcommand{roman(1992)}
\indentrel{3}\begin{verbatim}
   (5)  MCMXCII
                                     Type: RomanNumeral
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty5}
\begin{paste}{ugIntroNumbersPageEmpty5}{ugIntroNumbersPagePatch5}
\pastebutton{ugIntroNumbersPageEmpty5}{\showpaste}
\tab{5}\spadcommand{roman(1992)}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch6}
\begin{paste}{ugIntroNumbersPageFull6}{ugIntroNumbersPageEmpty6}
\pastebutton{ugIntroNumbersPageFull6}{\hidepaste}
\tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }}
\indentrel{3}\begin{verbatim}
        55739
   (6)  -----
         2520
                                Type: Fraction(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty6}
\begin{paste}{ugIntroNumbersPageEmpty6}{ugIntroNumbersPagePatch6}
\pastebutton{ugIntroNumbersPageEmpty6}{\showpaste}
\tab{5}\spadcommand{r := 10 + 9/2 + 8/3 + 7/4 + 6/5 + 5/6 + 4/7 + 3/8 + 2/9\bound{r }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch7}
\begin{paste}{ugIntroNumbersPageFull7}{ugIntroNumbersPageEmpty7}
\pastebutton{ugIntroNumbersPageFull7}{\hidepaste}
\tab{5}\spadcommand{map(factor,r)\free{r }}
\indentrel{3}\begin{verbatim}
        139 401
   (7)  -------
         3 2
        2 3 5 7
                      Type: Fraction(Factored(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty7}
\begin{paste}{ugIntroNumbersPageEmpty7}{ugIntroNumbersPagePatch7}
\pastebutton{ugIntroNumbersPageEmpty7}{\showpaste}
\tab{5}\spadcommand{map(factor,r)\free{r }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch8}
\begin{paste}{ugIntroNumbersPageFull8}{ugIntroNumbersPageEmpty8}
\pastebutton{ugIntroNumbersPageFull8}{\hidepaste}
\tab{5}\spadcommand{11@SingleInteger}
\indentrel{3}\begin{verbatim}
   (8)  11
                                    Type: SingleInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty8}
\begin{paste}{ugIntroNumbersPageEmpty8}{ugIntroNumbersPagePatch8}
\pastebutton{ugIntroNumbersPageEmpty8}{\showpaste}
\tab{5}\spadcommand{11@SingleInteger}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch9}
\begin{paste}{ugIntroNumbersPageFull9}{ugIntroNumbersPageEmpty9}
\pastebutton{ugIntroNumbersPageFull9}{\hidepaste}
\tab{5}\spadcommand{123.21@DoubleFloat}
\indentrel{3}\begin{verbatim}
   (9)  123.21000000000001
                                      Type: DoubleFloat
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty9}
\begin{paste}{ugIntroNumbersPageEmpty9}{ugIntroNumbersPagePatch9}
\pastebutton{ugIntroNumbersPageEmpty9}{\showpaste}
\tab{5}\spadcommand{123.21@DoubleFloat}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch10}
\begin{paste}{ugIntroNumbersPageFull10}{ugIntroNumbersPageEmpty10}
\pastebutton{ugIntroNumbersPageFull10}{\hidepaste}
\tab{5}\spadcommand{r :: Float\free{r }}
\indentrel{3}\begin{verbatim}
   (10)  22.1186507936_50793651
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty10}
\begin{paste}{ugIntroNumbersPageEmpty10}{ugIntroNumbersPagePatch10}
\pastebutton{ugIntroNumbersPageEmpty10}{\showpaste}
\tab{5}\spadcommand{r :: Float\free{r }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch11}
\begin{paste}{ugIntroNumbersPageFull11}{ugIntroNumbersPageEmpty11}
\pastebutton{ugIntroNumbersPageFull11}{\hidepaste}
\tab{5}\spadcommand{digits(22)\bound{fewerdigits }}
\indentrel{3}\begin{verbatim}
   (11)  20
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty11}
\begin{paste}{ugIntroNumbersPageEmpty11}{ugIntroNumbersPagePatch11}
\pastebutton{ugIntroNumbersPageEmpty11}{\showpaste}
\tab{5}\spadcommand{digits(22)\bound{fewerdigits }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch12}
\begin{paste}{ugIntroNumbersPageFull12}{ugIntroNumbersPageEmpty12}
\pastebutton{ugIntroNumbersPageFull12}{\hidepaste}
\tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }}
\indentrel{3}\begin{verbatim}
   (12)  26253741_2640768744.0
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty12}
\begin{paste}{ugIntroNumbersPageEmpty12}{ugIntroNumbersPagePatch12}
\pastebutton{ugIntroNumbersPageEmpty12}{\showpaste}
\tab{5}\spadcommand{exp(\%pi * sqrt 163.0)\free{fewerdigits }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch13}
\begin{paste}{ugIntroNumbersPageFull13}{ugIntroNumbersPageEmpty13}
\pastebutton{ugIntroNumbersPageFull13}{\hidepaste}
\tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }}
\indentrel{3}\begin{verbatim}
   (13)  26253741_2640768743.9999999999_9925007259_76
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty13}
\begin{paste}{ugIntroNumbersPageEmpty13}{ugIntroNumbersPagePatch13}
\pastebutton{ugIntroNumbersPageEmpty13}{\showpaste}
\tab{5}\spadcommand{digits(40); exp(\%pi * sqrt 163.0)\free{moredigits }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch14}
\begin{paste}{ugIntroNumbersPageFull14}{ugIntroNumbersPageEmpty14}
\pastebutton{ugIntroNumbersPageFull14}{\hidepaste}
\tab{5}\spadcommand{(2/3 + \%i)^3\bound{gaussint }}
\indentrel{3}\begin{verbatim}
           46   1
   (14)  - -- + - %i
           27   3
                       Type: Complex(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty14}
\begin{paste}{ugIntroNumbersPageEmpty14}{ugIntroNumbersPagePatch14}
\pastebutton{ugIntroNumbersPageEmpty14}{\showpaste}
\tab{5}\spadcommand{(2/3 + \%i)^3\bound{gaussint }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch15}
\begin{paste}{ugIntroNumbersPageFull15}{ugIntroNumbersPageEmpty15}
\pastebutton{ugIntroNumbersPageFull15}{\hidepaste}
\tab{5}\spadcommand{conjugate \%\free{gaussint }}
\indentrel{3}\begin{verbatim}
           46   1
   (15)  - -- - - %i
           27   3
                       Type: Complex(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty15}
\begin{paste}{ugIntroNumbersPageEmpty15}{ugIntroNumbersPagePatch15}
\pastebutton{ugIntroNumbersPageEmpty15}{\showpaste}
\tab{5}\spadcommand{conjugate \%\free{gaussint }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch16}
\begin{paste}{ugIntroNumbersPageFull16}{ugIntroNumbersPageEmpty16}
\pastebutton{ugIntroNumbersPageFull16}{\hidepaste}
\tab{5}\spadcommand{factor(89 - 23 * \%i)}
\indentrel{3}\begin{verbatim}
                           2          2
   (16)  - (1 + %i)(2 + %i) (3 + 2 %i)
                       Type: Factored(Complex(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty16}
\begin{paste}{ugIntroNumbersPageEmpty16}{ugIntroNumbersPagePatch16}
\pastebutton{ugIntroNumbersPageEmpty16}{\showpaste}
\tab{5}\spadcommand{factor(89 - 23 * \%i)}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch17}
\begin{paste}{ugIntroNumbersPageFull17}{ugIntroNumbersPageEmpty17}
\pastebutton{ugIntroNumbersPageFull17}{\hidepaste}
\tab{5}\spadcommand{exp(\%pi/4.0 * \%i)}
\indentrel{3}\begin{verbatim}
   (17)
     0.7071067811_8654752440_0844362104_8490392849
   + 
     0.7071067811_8654752440_0844362104_8490392848 %i
                                   Type: Complex(Float)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty17}
\begin{paste}{ugIntroNumbersPageEmpty17}{ugIntroNumbersPagePatch17}
\pastebutton{ugIntroNumbersPageEmpty17}{\showpaste}
\tab{5}\spadcommand{exp(\%pi/4.0 * \%i)}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch18}
\begin{paste}{ugIntroNumbersPageFull18}{ugIntroNumbersPageEmpty18}
\pastebutton{ugIntroNumbersPageFull18}{\hidepaste}
\tab{5}\spadcommand{decimal(1/352)}
\indentrel{3}\begin{verbatim}
                __
   (18)  0.0028409
                                 Type: DecimalExpansion
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty18}
\begin{paste}{ugIntroNumbersPageEmpty18}{ugIntroNumbersPagePatch18}
\pastebutton{ugIntroNumbersPageEmpty18}{\showpaste}
\tab{5}\spadcommand{decimal(1/352)}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch19}
\begin{paste}{ugIntroNumbersPageFull19}{ugIntroNumbersPageEmpty19}
\pastebutton{ugIntroNumbersPageFull19}{\hidepaste}
\tab{5}\spadcommand{continuedFraction(6543/210)}
\indentrel{3}\begin{verbatim}
                1 |     1 |     1 |     1 |
   (19)  31 + +---+ + +---+ + +---+ + +---+
              | 6     | 2     | 1     | 3
                       Type: ContinuedFraction(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty19}
\begin{paste}{ugIntroNumbersPageEmpty19}{ugIntroNumbersPagePatch19}
\pastebutton{ugIntroNumbersPageEmpty19}{\showpaste}
\tab{5}\spadcommand{continuedFraction(6543/210)}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch20}
\begin{paste}{ugIntroNumbersPageFull20}{ugIntroNumbersPageEmpty20}
\pastebutton{ugIntroNumbersPageFull20}{\hidepaste}
\tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }}
\indentrel{3}\begin{verbatim}
         159   23   12   1
   (20)  --- - -- - -- + -
           8    4    2   7
          2    3    5
                         Type: PartialFraction(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty20}
\begin{paste}{ugIntroNumbersPageEmpty20}{ugIntroNumbersPagePatch20}
\pastebutton{ugIntroNumbersPageEmpty20}{\showpaste}
\tab{5}\spadcommand{partialFraction(1,factorial(10))\bound{partfrac }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch21}
\begin{paste}{ugIntroNumbersPageFull21}{ugIntroNumbersPageEmpty21}
\pastebutton{ugIntroNumbersPageFull21}{\hidepaste}
\tab{5}\spadcommand{padicFraction(\%)\free{partfrac }}
\indentrel{3}\begin{verbatim}
   (21)
   1    1    1    1    1    1    2    1    2   2    2   1
   - + -- + -- + -- + -- + -- - -- - -- - -- - - - -- + -
   2    4    5    6    7    8    2    3    4   5    2   7
       2    2    2    2    2    3    3    3        5
                         Type: PartialFraction(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty21}
\begin{paste}{ugIntroNumbersPageEmpty21}{ugIntroNumbersPagePatch21}
\pastebutton{ugIntroNumbersPageEmpty21}{\showpaste}
\tab{5}\spadcommand{padicFraction(\%)\free{partfrac }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch22}
\begin{paste}{ugIntroNumbersPageFull22}{ugIntroNumbersPageEmpty22}
\pastebutton{ugIntroNumbersPageFull22}{\hidepaste}
\tab{5}\spadcommand{radix(4/7, 8)\bound{rad }}
\indentrel{3}\begin{verbatim}
           _
   (22)  0.4
                                Type: RadixExpansion(8)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty22}
\begin{paste}{ugIntroNumbersPageEmpty22}{ugIntroNumbersPagePatch22}
\pastebutton{ugIntroNumbersPageEmpty22}{\showpaste}
\tab{5}\spadcommand{radix(4/7, 8)\bound{rad }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch23}
\begin{paste}{ugIntroNumbersPageFull23}{ugIntroNumbersPageEmpty23}
\pastebutton{ugIntroNumbersPageFull23}{\hidepaste}
\tab{5}\spadcommand{\% + 2/3*\%i\free{rad }}
\indentrel{3}\begin{verbatim}
         4   2
   (23)  - + - %i
         7   3
                       Type: Complex(Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty23}
\begin{paste}{ugIntroNumbersPageEmpty23}{ugIntroNumbersPagePatch23}
\pastebutton{ugIntroNumbersPageEmpty23}{\showpaste}
\tab{5}\spadcommand{\% + 2/3*\%i\free{rad }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch24}
\begin{paste}{ugIntroNumbersPageFull24}{ugIntroNumbersPageEmpty24}
\pastebutton{ugIntroNumbersPageFull24}{\hidepaste}
\tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)^(1/3)}
\indentrel{3}\begin{verbatim}
          +-----------+
         3|    +-+
   (24)  \|14 \|7  + 5
                                  Type: AlgebraicNumber
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty24}
\begin{paste}{ugIntroNumbersPageEmpty24}{ugIntroNumbersPagePatch24}
\pastebutton{ugIntroNumbersPageEmpty24}{\showpaste}
\tab{5}\spadcommand{(5 + sqrt 63 + sqrt 847)^(1/3)}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch25}
\begin{paste}{ugIntroNumbersPageFull25}{ugIntroNumbersPageEmpty25}
\pastebutton{ugIntroNumbersPageFull25}{\hidepaste}
\tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }}
\indentrel{3}\begin{verbatim}
   (25)  5
                                    Type: PrimeField(7)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty25}
\begin{paste}{ugIntroNumbersPageEmpty25}{ugIntroNumbersPagePatch25}
\pastebutton{ugIntroNumbersPageEmpty25}{\showpaste}
\tab{5}\spadcommand{x : PrimeField 7 := 5\bound{x }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch26}
\begin{paste}{ugIntroNumbersPageFull26}{ugIntroNumbersPageEmpty26}
\pastebutton{ugIntroNumbersPageFull26}{\hidepaste}
\tab{5}\spadcommand{x^3\free{x }}
\indentrel{3}\begin{verbatim}
   (26)  6
                                    Type: PrimeField(7)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty26}
\begin{paste}{ugIntroNumbersPageEmpty26}{ugIntroNumbersPagePatch26}
\pastebutton{ugIntroNumbersPageEmpty26}{\showpaste}
\tab{5}\spadcommand{x^3\free{x }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch27}
\begin{paste}{ugIntroNumbersPageFull27}{ugIntroNumbersPageEmpty27}
\pastebutton{ugIntroNumbersPageFull27}{\hidepaste}
\tab{5}\spadcommand{1/x\free{x }}
\indentrel{3}\begin{verbatim}
   (27)  3
                                    Type: PrimeField(7)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty27}
\begin{paste}{ugIntroNumbersPageEmpty27}{ugIntroNumbersPagePatch27}
\pastebutton{ugIntroNumbersPageEmpty27}{\showpaste}
\tab{5}\spadcommand{1/x\free{x }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch28}
\begin{paste}{ugIntroNumbersPageFull28}{ugIntroNumbersPageEmpty28}
\pastebutton{ugIntroNumbersPageFull28}{\hidepaste}
\tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }}
\indentrel{3}\begin{verbatim}
   (28)  5
                                    Type: IntegerMod(6)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty28}
\begin{paste}{ugIntroNumbersPageEmpty28}{ugIntroNumbersPagePatch28}
\pastebutton{ugIntroNumbersPageEmpty28}{\showpaste}
\tab{5}\spadcommand{y : IntegerMod 6 := 5\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch29}
\begin{paste}{ugIntroNumbersPageFull29}{ugIntroNumbersPageEmpty29}
\pastebutton{ugIntroNumbersPageFull29}{\hidepaste}
\tab{5}\spadcommand{y^3\free{y }}
\indentrel{3}\begin{verbatim}
   (29)  5
                                    Type: IntegerMod(6)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty29}
\begin{paste}{ugIntroNumbersPageEmpty29}{ugIntroNumbersPagePatch29}
\pastebutton{ugIntroNumbersPageEmpty29}{\showpaste}
\tab{5}\spadcommand{y^3\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch30}
\begin{paste}{ugIntroNumbersPageFull30}{ugIntroNumbersPageEmpty30}
\pastebutton{ugIntroNumbersPageFull30}{\hidepaste}
\tab{5}\spadcommand{1/y\free{y }}
\indentrel{3}\begin{verbatim}
error
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty30}
\begin{paste}{ugIntroNumbersPageEmpty30}{ugIntroNumbersPagePatch30}
\pastebutton{ugIntroNumbersPageEmpty30}{\showpaste}
\tab{5}\spadcommand{1/y\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch31}
\begin{paste}{ugIntroNumbersPageFull31}{ugIntroNumbersPageEmpty31}
\pastebutton{ugIntroNumbersPageFull31}{\hidepaste}
\tab{5}\spadcommand{a := rootOf(a^5 + a^3 + a^2 + 3,a)\bound{a }}
\indentrel{3}\begin{verbatim}
   (30)  a
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty31}
\begin{paste}{ugIntroNumbersPageEmpty31}{ugIntroNumbersPagePatch31}
\pastebutton{ugIntroNumbersPageEmpty31}{\showpaste}
\tab{5}\spadcommand{a := rootOf(a^5 + a^3 + a^2 + 3,a)\bound{a }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch32}
\begin{paste}{ugIntroNumbersPageFull32}{ugIntroNumbersPageEmpty32}
\pastebutton{ugIntroNumbersPageFull32}{\hidepaste}
\tab{5}\spadcommand{(a + 1)^10\free{a }}
\indentrel{3}\begin{verbatim}
               4        3        2
   (31)  - 85 a  - 264 a  - 378 a  - 458 a - 287
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty32}
\begin{paste}{ugIntroNumbersPageEmpty32}{ugIntroNumbersPagePatch32}
\pastebutton{ugIntroNumbersPageEmpty32}{\showpaste}
\tab{5}\spadcommand{(a + 1)^10\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch33}
\begin{paste}{ugIntroNumbersPageFull33}{ugIntroNumbersPageEmpty33}
\pastebutton{ugIntroNumbersPageFull33}{\hidepaste}
\tab{5}\spadcommand{b := rootOf(b^4 + a,b)\bound{b }\free{a }}
\indentrel{3}\begin{verbatim}
   (32)  b
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty33}
\begin{paste}{ugIntroNumbersPageEmpty33}{ugIntroNumbersPagePatch33}
\pastebutton{ugIntroNumbersPageEmpty33}{\showpaste}
\tab{5}\spadcommand{b := rootOf(b^4 + a,b)\bound{b }\free{a }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch34}
\begin{paste}{ugIntroNumbersPageFull34}{ugIntroNumbersPageEmpty34}
\pastebutton{ugIntroNumbersPageFull34}{\hidepaste}
\tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }}
\indentrel{3}\begin{verbatim}
           2
   (33)  -----
         b - 1
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty34}
\begin{paste}{ugIntroNumbersPageEmpty34}{ugIntroNumbersPagePatch34}
\pastebutton{ugIntroNumbersPageEmpty34}{\showpaste}
\tab{5}\spadcommand{2/(b - 1)\free{b }\bound{check }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch35}
\begin{paste}{ugIntroNumbersPageFull35}{ugIntroNumbersPageEmpty35}
\pastebutton{ugIntroNumbersPageFull35}{\hidepaste}
\tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }}
\indentrel{3}\begin{verbatim}
   (34)
       4    3      2          3
     (a  - a  + 2 a  - a + 1)b
   + 
       4    3      2          2
     (a  - a  + 2 a  - a + 1)b
   + 
       4    3      2              4    3      2
     (a  - a  + 2 a  - a + 1)b + a  - a  + 2 a  - a + 1
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty35}
\begin{paste}{ugIntroNumbersPageEmpty35}{ugIntroNumbersPagePatch35}
\pastebutton{ugIntroNumbersPageEmpty35}{\showpaste}
\tab{5}\spadcommand{ratDenom(\%)\free{check }\bound{check1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch36}
\begin{paste}{ugIntroNumbersPageFull36}{ugIntroNumbersPageEmpty36}
\pastebutton{ugIntroNumbersPageFull36}{\hidepaste}
\tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }}
\indentrel{3}\begin{verbatim}
   (35)
         4    3      2          3
       (a  - a  + 2 a  - a + 1)b
     + 
         4    3      2          2
       (a  - a  + 2 a  - a + 1)b
     + 
         4    3      2              4    3      2
       (a  - a  + 2 a  - a + 1)b + a  - a  + 2 a  - a + 3
  /
         4    3      2          3
       (a  - a  + 2 a  - a + 1)b
     + 
         4    3      2          2
       (a  - a  + 2 a  - a + 1)b
     + 
         4    3      2              4    3      2
       (a  - a  + 2 a  - a + 1)b + a  - a  + 2 a  - a + 1
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty36}
\begin{paste}{ugIntroNumbersPageEmpty36}{ugIntroNumbersPagePatch36}
\pastebutton{ugIntroNumbersPageEmpty36}{\showpaste}
\tab{5}\spadcommand{2/\%+1\free{check1 }\bound{check2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch37}
\begin{paste}{ugIntroNumbersPageFull37}{ugIntroNumbersPageEmpty37}
\pastebutton{ugIntroNumbersPageFull37}{\hidepaste}
\tab{5}\spadcommand{ratDenom(\%)\free{check2 }}
\indentrel{3}\begin{verbatim}
   (36)  b
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty37}
\begin{paste}{ugIntroNumbersPageEmpty37}{ugIntroNumbersPagePatch37}
\pastebutton{ugIntroNumbersPageEmpty37}{\showpaste}
\tab{5}\spadcommand{ratDenom(\%)\free{check2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPagePatch38}
\begin{paste}{ugIntroNumbersPageFull38}{ugIntroNumbersPageEmpty38}
\pastebutton{ugIntroNumbersPageFull38}{\hidepaste}
\tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)}
\indentrel{3}\begin{verbatim}
   (37)  - 8 i + 16 j - 8 k
                              Type: Quaternion(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroNumbersPageEmpty38}
\begin{paste}{ugIntroNumbersPageEmpty38}{ugIntroNumbersPagePatch38}
\pastebutton{ugIntroNumbersPageEmpty38}{\showpaste}
\tab{5}\spadcommand{q:=quatern(1,2,3,4)*quatern(5,6,7,8) - quatern(5,6,7,8)*quatern(1,2,3,4)}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch1}
\begin{paste}{ugIntroDiffEqnsPageFull1}{ugIntroDiffEqnsPageEmpty1}
\pastebutton{ugIntroDiffEqnsPageFull1}{\hidepaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\indentrel{3}\begin{verbatim}
   (1)  y
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty1}
\begin{paste}{ugIntroDiffEqnsPageEmpty1}{ugIntroDiffEqnsPagePatch1}
\pastebutton{ugIntroDiffEqnsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{y := operator 'y\bound{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch2}
\begin{paste}{ugIntroDiffEqnsPageFull2}{ugIntroDiffEqnsPageEmpty2}
\pastebutton{ugIntroDiffEqnsPageFull2}{\hidepaste}
\tab{5}\spadcommand{deq := x^3 * D(y x, x, 3) + x^2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x^4\bound{e3 }\free{y }}
\indentrel{3}\begin{verbatim}
         3 ,,,       2 ,,          ,                  4
   (2)  x y   (x) + x y  (x) - 2 xy (x) + 2 y(x) = 2 x

                    Type: Equation(Expression(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty2}
\begin{paste}{ugIntroDiffEqnsPageEmpty2}{ugIntroDiffEqnsPagePatch2}
\pastebutton{ugIntroDiffEqnsPageEmpty2}{\showpaste}
\tab{5}\spadcommand{deq := x^3 * D(y x, x, 3) + x^2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x^4\bound{e3 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch3}
\begin{paste}{ugIntroDiffEqnsPageFull3}{ugIntroDiffEqnsPageEmpty3}
\pastebutton{ugIntroDiffEqnsPageFull3}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
\indentrel{3}\begin{verbatim}
   (3)
                  5       3       2
                 x  - 10 x  + 20 x  + 4
   [particular = ----------------------,
                          15 x
                3      2       3       3      2
             2 x  - 3 x  + 1  x  - 1  x  - 3 x  - 1
    basis = [---------------, ------, -------------]]
                    x            x          x
Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty3}
\begin{paste}{ugIntroDiffEqnsPageEmpty3}{ugIntroDiffEqnsPagePatch3}
\pastebutton{ugIntroDiffEqnsPageEmpty3}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch4}
\begin{paste}{ugIntroDiffEqnsPageFull4}{ugIntroDiffEqnsPageEmpty4}
\pastebutton{ugIntroDiffEqnsPageFull4}{\hidepaste}
\tab{5}\spadcommand{deq := (x^2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
\indentrel{3}\begin{verbatim}
          2      ,,          ,
   (4)  (x  + 1)y  (x) + 3 xy (x) + y(x) = 0

                    Type: Equation(Expression(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty4}
\begin{paste}{ugIntroDiffEqnsPageEmpty4}{ugIntroDiffEqnsPagePatch4}
\pastebutton{ugIntroDiffEqnsPageEmpty4}{\showpaste}
\tab{5}\spadcommand{deq := (x^2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch5}
\begin{paste}{ugIntroDiffEqnsPageFull5}{ugIntroDiffEqnsPageEmpty5}
\pastebutton{ugIntroDiffEqnsPageFull5}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
\indentrel{3}\begin{verbatim}
   (5)
   [particular = 0,
                             +------+
                             | 2
                 1      log(\|x  + 1  - x)
    basis = [---------, ------------------]]
              +------+        +------+
              | 2             | 2
             \|x  + 1        \|x  + 1
Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty5}
\begin{paste}{ugIntroDiffEqnsPageEmpty5}{ugIntroDiffEqnsPagePatch5}
\pastebutton{ugIntroDiffEqnsPageEmpty5}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch6}
\begin{paste}{ugIntroDiffEqnsPageFull6}{ugIntroDiffEqnsPageEmpty6}
\pastebutton{ugIntroDiffEqnsPageFull6}{\hidepaste}
\tab{5}\spadcommand{eq := 2*x^3 * D(y x,x,2) + 3*x^2 * D(y x,x) - 2 * y x\bound{eq }\free{y }}
\indentrel{3}\begin{verbatim}
           3 ,,         2 ,
   (6)  2 x y  (x) + 3 x y (x) - 2 y(x)

                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty6}
\begin{paste}{ugIntroDiffEqnsPageEmpty6}{ugIntroDiffEqnsPagePatch6}
\pastebutton{ugIntroDiffEqnsPageEmpty6}{\showpaste}
\tab{5}\spadcommand{eq := 2*x^3 * D(y x,x,2) + 3*x^2 * D(y x,x) - 2 * y x\bound{eq }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch7}
\begin{paste}{ugIntroDiffEqnsPageFull7}{ugIntroDiffEqnsPageEmpty7}
\pastebutton{ugIntroDiffEqnsPageFull7}{\hidepaste}
\tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }}
\indentrel{3}\begin{verbatim}
               2       2
           - ----    ----
              +-+     +-+
             \|x     \|x
   (7)  [%e      , %e    ]
                        Type: List(Expression(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty7}
\begin{paste}{ugIntroDiffEqnsPageEmpty7}{ugIntroDiffEqnsPagePatch7}
\pastebutton{ugIntroDiffEqnsPageEmpty7}{\showpaste}
\tab{5}\spadcommand{solve(eq,y,x).basis\free{eq }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch8}
\begin{paste}{ugIntroDiffEqnsPageFull8}{ugIntroDiffEqnsPageEmpty8}
\pastebutton{ugIntroDiffEqnsPageFull8}{\hidepaste}
\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
\indentrel{3}\begin{verbatim}
         ,             y(x)
   (8)  y (x) = -----------------
                y(x)log(y(x)) + x
                    Type: Equation(Expression(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty8}
\begin{paste}{ugIntroDiffEqnsPageEmpty8}{ugIntroDiffEqnsPagePatch8}
\pastebutton{ugIntroDiffEqnsPageEmpty8}{\showpaste}
\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch9}
\begin{paste}{ugIntroDiffEqnsPageFull9}{ugIntroDiffEqnsPageEmpty9}
\pastebutton{ugIntroDiffEqnsPageFull9}{\hidepaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
\indentrel{3}\begin{verbatim}
                     2
        y(x)log(y(x))  - 2 x
   (9)  --------------------
               2 y(x)
                   Type: Union(Expression(Integer),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty9}
\begin{paste}{ugIntroDiffEqnsPageEmpty9}{ugIntroDiffEqnsPagePatch9}
\pastebutton{ugIntroDiffEqnsPageEmpty9}{\showpaste}
\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch10}
\begin{paste}{ugIntroDiffEqnsPageFull10}{ugIntroDiffEqnsPageEmpty10}
\pastebutton{ugIntroDiffEqnsPageFull10}{\hidepaste}
\tab{5}\spadcommand{x := operator 'x\bound{x }}
\indentrel{3}\begin{verbatim}
   (10)  x
                                    Type: BasicOperator
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty10}
\begin{paste}{ugIntroDiffEqnsPageEmpty10}{ugIntroDiffEqnsPagePatch10}
\pastebutton{ugIntroDiffEqnsPageEmpty10}{\showpaste}
\tab{5}\spadcommand{x := operator 'x\bound{x }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch11}
\begin{paste}{ugIntroDiffEqnsPageFull11}{ugIntroDiffEqnsPageEmpty11}
\pastebutton{ugIntroDiffEqnsPageFull11}{\hidepaste}
\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)^2\free{x }\free{y }\bound{eq1 }}
\indentrel{3}\begin{verbatim}
          ,          2
   (11)  x (t) = x(t)  + 1

                    Type: Equation(Expression(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty11}
\begin{paste}{ugIntroDiffEqnsPageEmpty11}{ugIntroDiffEqnsPagePatch11}
\pastebutton{ugIntroDiffEqnsPageEmpty11}{\showpaste}
\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)^2\free{x }\free{y }\bound{eq1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch12}
\begin{paste}{ugIntroDiffEqnsPageFull12}{ugIntroDiffEqnsPageEmpty12}
\pastebutton{ugIntroDiffEqnsPageFull12}{\hidepaste}
\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
\indentrel{3}\begin{verbatim}
          ,
   (12)  y (t) = x(t)y(t)

                    Type: Equation(Expression(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty12}
\begin{paste}{ugIntroDiffEqnsPageEmpty12}{ugIntroDiffEqnsPagePatch12}
\pastebutton{ugIntroDiffEqnsPageEmpty12}{\showpaste}
\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPagePatch13}
\begin{paste}{ugIntroDiffEqnsPageFull13}{ugIntroDiffEqnsPageEmpty13}
\pastebutton{ugIntroDiffEqnsPageFull13}{\hidepaste}
\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }}
\indentrel{3}\begin{verbatim}
   (13)
        1  3    2  5    17  7    62   9      11
   [t + - t  + -- t  + --- t  + ---- t  + O(t  ),
        3      15      315      2835

           1  2    5  4    61  6    277  8    50521   10
       1 + - t  + -- t  + --- t  + ---- t  + ------- t
           2      24      720      8064      3628800
     + 
          11
       O(t  )
     ]
Type: List(UnivariateTaylorSeries(Expression(Integer),t,0))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroDiffEqnsPageEmpty13}
\begin{paste}{ugIntroDiffEqnsPageEmpty13}{ugIntroDiffEqnsPagePatch13}
\pastebutton{ugIntroDiffEqnsPageEmpty13}{\showpaste}
\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroGraphicsPagePatch1}
\begin{paste}{ugIntroGraphicsPageFull1}{ugIntroGraphicsPageEmpty1}
\pastebutton{ugIntroGraphicsPageFull1}{\hidepaste}
\tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)}
\center{\unixcommand{\inputimage{\env{AXIOM}/share/viewports/ugIntroGraphicsPage1.VIEW/image}}{\env{AXIOM}/bin/viewAlone\space{1} \env{AXIOM}/share/viewports/ugIntroGraphicsPage1}}
\end{paste}\end{patch}

\begin{patch}{ugIntroGraphicsPageEmpty1}
\begin{paste}{ugIntroGraphicsPageEmpty1}{ugIntroGraphicsPagePatch1}
\pastebutton{ugIntroGraphicsPageEmpty1}{\showpaste}
\tab{5}\spadgraph{draw(cos(5*t/8), t=0..16*\%pi, coordinates==polar)}
\end{paste}\end{patch}

\begin{patch}{ugIntroGraphicsPagePatch2}
\begin{paste}{ugIntroGraphicsPageFull2}{ugIntroGraphicsPageEmpty2}
\pastebutton{ugIntroGraphicsPageFull2}{\hidepaste}
\tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))}
\center{\unixcommand{\inputimage{\env{AXIOM}/share/viewports/ugIntroGraphicsPage2.VIEW/image}}{\env{AXIOM}/bin/viewAlone\space{1} \env{AXIOM}/share/viewports/ugIntroGraphicsPage2}}
\end{paste}\end{patch}

\begin{patch}{ugIntroGraphicsPageEmpty2}
\begin{paste}{ugIntroGraphicsPageEmpty2}{ugIntroGraphicsPagePatch2}
\pastebutton{ugIntroGraphicsPageEmpty2}{\showpaste}
\tab{5}\spadgraph{draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction == (x,y) +-> argument atan complex(x,y))}
\end{paste}\end{patch}

\begin{patch}{ugIntroCommentsPagePatch1}
\begin{paste}{ugIntroCommentsPageFull1}{ugIntroCommentsPageEmpty1}
\pastebutton{ugIntroCommentsPageFull1}{\hidepaste}
\tab{5}\spadcommand{2 + 3 -- this is rather simple, no?}
\indentrel{3}\begin{verbatim}
   (1)  5
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroCommentsPageEmpty1}
\begin{paste}{ugIntroCommentsPageEmpty1}{ugIntroCommentsPagePatch1}
\pastebutton{ugIntroCommentsPageEmpty1}{\showpaste}
\tab{5}\spadcommand{2 + 3 -- this is rather simple, no?}
\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPagePatch1}
\begin{paste}{ugIntroSolutionPageFull1}{ugIntroSolutionPageEmpty1}
\pastebutton{ugIntroSolutionPageFull1}{\hidepaste}
\tab{5}\spadcommand{S(t) == [x^2-2*y^2 - t,x*y-y-5*x + 5]\bound{S1 }}
\indentrel{3}\begin{verbatim}
                                             Type: Void
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPageEmpty1}
\begin{paste}{ugIntroSolutionPageEmpty1}{ugIntroSolutionPagePatch1}
\pastebutton{ugIntroSolutionPageEmpty1}{\showpaste}
\tab{5}\spadcommand{S(t) == [x^2-2*y^2 - t,x*y-y-5*x + 5]\bound{S1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPagePatch2}
\begin{paste}{ugIntroSolutionPageFull2}{ugIntroSolutionPageEmpty2}
\pastebutton{ugIntroSolutionPageFull2}{\hidepaste}
\tab{5}\spadcommand{solve(S(19),1/10^20)\free{S1 }}
\indentrel{3}\begin{verbatim}
   (2)
                  80336736493669365924189585
   [[y = 5, x = - --------------------------],
                   9671406556917033397649408
                80336736493669365924189585
    [y = 5, x = --------------------------]]
                 9671406556917033397649408
Type: List(List(Equation(Polynomial(Fraction(Integer)))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPageEmpty2}
\begin{paste}{ugIntroSolutionPageEmpty2}{ugIntroSolutionPagePatch2}
\pastebutton{ugIntroSolutionPageEmpty2}{\showpaste}
\tab{5}\spadcommand{solve(S(19),1/10^20)\free{S1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPagePatch3}
\begin{paste}{ugIntroSolutionPageFull3}{ugIntroSolutionPageEmpty3}
\pastebutton{ugIntroSolutionPageFull3}{\hidepaste}
\tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }}
\indentrel{3}\begin{verbatim}
   (3)
   [
     [y = 5.0,

         x
       = 
        8.3066238629_1807485258_4262744905_6951556981_516
        9148184_0582865006_639146088
       ]
     ,

     [y = 5.0,
      x = - 8.3066238629_1807485258_4262744905_695155698]
     ,
    [y = - 3.0 %i, x = 1.0], [y = 3.0 %i, x = 1.0]]
 Type: List(List(Equation(Polynomial(Complex(Float)))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPageEmpty3}
\begin{paste}{ugIntroSolutionPageEmpty3}{ugIntroSolutionPagePatch3}
\pastebutton{ugIntroSolutionPageEmpty3}{\showpaste}
\tab{5}\spadcommand{complexSolve(S(19),10.e-20)\free{S1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPagePatch4}
\begin{paste}{ugIntroSolutionPageFull4}{ugIntroSolutionPageEmpty4}
\pastebutton{ugIntroSolutionPageFull4}{\hidepaste}
\tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }}
\indentrel{3}\begin{verbatim}
   (4)
            +------+                +------+
   [[x = - \|a + 50 , y = 5], [x = \|a + 50 , y = 5],
                 +-------+                  +-------+
                \|- a + 1                  \|- a + 1
    [x = 1, y = ----------], [x = 1, y = - ----------]]
                    +-+                        +-+
                   \|2                        \|2
        Type: List(List(Equation(Expression(Integer))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPageEmpty4}
\begin{paste}{ugIntroSolutionPageEmpty4}{ugIntroSolutionPagePatch4}
\pastebutton{ugIntroSolutionPageEmpty4}{\showpaste}
\tab{5}\spadcommand{radicalSolve(S(a),[x,y])\free{S1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPagePatch5}
\begin{paste}{ugIntroSolutionPageFull5}{ugIntroSolutionPageEmpty5}
\pastebutton{ugIntroSolutionPageFull5}{\hidepaste}
\tab{5}\spadcommand{eqns := [x^2 - y + z,x^2*z + x^4 - b*y, y^2 *z - a - b*x]\bound{e }}
\indentrel{3}\begin{verbatim}
                  2   2           4   2
   (5)  [z - y + x , x z - b y + x , y z - b x - a]
                        Type: List(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPageEmpty5}
\begin{paste}{ugIntroSolutionPageEmpty5}{ugIntroSolutionPagePatch5}
\pastebutton{ugIntroSolutionPageEmpty5}{\showpaste}
\tab{5}\spadcommand{eqns := [x^2 - y + z,x^2*z + x^4 - b*y, y^2 *z - a - b*x]\bound{e }}
\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPagePatch6}
\begin{paste}{ugIntroSolutionPageFull6}{ugIntroSolutionPageEmpty6}
\pastebutton{ugIntroSolutionPageFull6}{\hidepaste}
\tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }}
\indentrel{3}\begin{verbatim}
   (6)
                            2
           a               a
   [[x = - -, y = 0, z = - --],
           b                2
                           b

           3        2    2
          z  + 2 b z  + b z - a
     [x = ---------------------, y = z + b,
                    b

            6        5      2 4       3        3
           z  + 4 b z  + 6 b z  + (4 b  - 2 a)z
         + 
             4          2        2     3    2
           (b  - 4 a b)z  - 2 a b z - b  + a
       = 
         0
       ]
     ]
Type: List(List(Equation(Fraction(Polynomial(Integer)))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroSolutionPageEmpty6}
\begin{paste}{ugIntroSolutionPageEmpty6}{ugIntroSolutionPagePatch6}
\pastebutton{ugIntroSolutionPageEmpty6}{\showpaste}
\tab{5}\spadcommand{solve(eqns,[x,y,z])\free{e }}
\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPagePatch1}
\begin{paste}{ugIntroPreviousPageFull1}{ugIntroPreviousPageEmpty1}
\pastebutton{ugIntroPreviousPageFull1}{\hidepaste}
\tab{5}\spadcommand{10 ^ 10\bound{prev }}
\indentrel{3}\begin{verbatim}
   (1)  10000000000
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPageEmpty1}
\begin{paste}{ugIntroPreviousPageEmpty1}{ugIntroPreviousPagePatch1}
\pastebutton{ugIntroPreviousPageEmpty1}{\showpaste}
\tab{5}\spadcommand{10 ^ 10\bound{prev }}
\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPagePatch2}
\begin{paste}{ugIntroPreviousPageFull2}{ugIntroPreviousPageEmpty2}
\pastebutton{ugIntroPreviousPageFull2}{\hidepaste}
\tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }}
\indentrel{3}\begin{verbatim}
   (2)  9999999999
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPageEmpty2}
\begin{paste}{ugIntroPreviousPageEmpty2}{ugIntroPreviousPagePatch2}
\pastebutton{ugIntroPreviousPageEmpty2}{\showpaste}
\tab{5}\spadcommand{\% - 1\free{prev }\bound{prev1 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPagePatch3}
\begin{paste}{ugIntroPreviousPageFull3}{ugIntroPreviousPageEmpty3}
\pastebutton{ugIntroPreviousPageFull3}{\hidepaste}
\tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }}
\indentrel{3}\begin{verbatim}
   (3)  9999999999
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPageEmpty3}
\begin{paste}{ugIntroPreviousPageEmpty3}{ugIntroPreviousPagePatch3}
\pastebutton{ugIntroPreviousPageEmpty3}{\showpaste}
\tab{5}\spadcommand{\%\%(-1)\free{prev1 }\bound{prev2 }}
\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPagePatch4}
\begin{paste}{ugIntroPreviousPageFull4}{ugIntroPreviousPageEmpty4}
\pastebutton{ugIntroPreviousPageFull4}{\hidepaste}
\tab{5}\spadcommand{\%\%(1)\free{prev2 }}
\indentrel{3}\begin{verbatim}
   (4)  10000000000
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ugIntroPreviousPageEmpty4}
\begin{paste}{ugIntroPreviousPageEmpty4}{ugIntroPreviousPagePatch4}
\pastebutton{ugIntroPreviousPageEmpty4}{\showpaste}
\tab{5}\spadcommand{\%\%(1)\free{prev2 }}
\end{paste}\end{patch}

