\begin{patch}{XPolynomialRingXmpPagePatch1}
\begin{paste}{XPolynomialRingXmpPageFull1}{XPolynomialRingXmpPageEmpty1}
\pastebutton{XPolynomialRingXmpPageFull1}{\hidepaste}
\tab{5}\spadcommand{Word := FreeMonoid(Symbol)\bound{Word }}
\indentrel{3}\begin{verbatim}
   (1)  FreeMonoid(Symbol)
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty1}
\begin{paste}{XPolynomialRingXmpPageEmpty1}{XPolynomialRingXmpPagePatch1}
\pastebutton{XPolynomialRingXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{Word := FreeMonoid(Symbol)\bound{Word }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch2}
\begin{paste}{XPolynomialRingXmpPageFull2}{XPolynomialRingXmpPageEmpty2}
\pastebutton{XPolynomialRingXmpPageFull2}{\hidepaste}
\tab{5}\spadcommand{poly:= XPR(Integer,Word)\free{Word }\bound{poly }}
\indentrel{3}\begin{verbatim}
   (2)  XPolynomialRing(Integer,FreeMonoid(Symbol))
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty2}
\begin{paste}{XPolynomialRingXmpPageEmpty2}{XPolynomialRingXmpPagePatch2}
\pastebutton{XPolynomialRingXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{poly:= XPR(Integer,Word)\free{Word }\bound{poly }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch3}
\begin{paste}{XPolynomialRingXmpPageFull3}{XPolynomialRingXmpPageEmpty3}
\pastebutton{XPolynomialRingXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{p:poly := 2 * x - 3 * y + 1\free{poly }\bound{p }}
\indentrel{3}\begin{verbatim}
   (3)  1 + 2 x - 3 y
      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty3}
\begin{paste}{XPolynomialRingXmpPageEmpty3}{XPolynomialRingXmpPagePatch3}
\pastebutton{XPolynomialRingXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{p:poly := 2 * x - 3 * y + 1\free{poly }\bound{p }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch4}
\begin{paste}{XPolynomialRingXmpPageFull4}{XPolynomialRingXmpPageEmpty4}
\pastebutton{XPolynomialRingXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{q:poly := 2 * x + 1\free{poly }\bound{q }}
\indentrel{3}\begin{verbatim}
   (4)  1 + 2 x
      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty4}
\begin{paste}{XPolynomialRingXmpPageEmpty4}{XPolynomialRingXmpPagePatch4}
\pastebutton{XPolynomialRingXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{q:poly := 2 * x + 1\free{poly }\bound{q }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch5}
\begin{paste}{XPolynomialRingXmpPageFull5}{XPolynomialRingXmpPageEmpty5}
\pastebutton{XPolynomialRingXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{p + q\free{p }\free{q }}
\indentrel{3}\begin{verbatim}
   (5)  2 + 4 x - 3 y
      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty5}
\begin{paste}{XPolynomialRingXmpPageEmpty5}{XPolynomialRingXmpPagePatch5}
\pastebutton{XPolynomialRingXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{p + q\free{p }\free{q }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch6}
\begin{paste}{XPolynomialRingXmpPageFull6}{XPolynomialRingXmpPageEmpty6}
\pastebutton{XPolynomialRingXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{p * q\free{p }\free{q }}
\indentrel{3}\begin{verbatim}
                           2
   (6)  1 + 4 x - 3 y + 4 x  - 6 y x
      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty6}
\begin{paste}{XPolynomialRingXmpPageEmpty6}{XPolynomialRingXmpPagePatch6}
\pastebutton{XPolynomialRingXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{p * q\free{p }\free{q }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch7}
\begin{paste}{XPolynomialRingXmpPageFull7}{XPolynomialRingXmpPageEmpty7}
\pastebutton{XPolynomialRingXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{(p +q)^2 -p^2 -q^2 - 2*p*q\free{p }\free{q }}
\indentrel{3}\begin{verbatim}
   (7)  - 6 x y + 6 y x
      Type: XPolynomialRing(Integer,FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty7}
\begin{paste}{XPolynomialRingXmpPageEmpty7}{XPolynomialRingXmpPagePatch7}
\pastebutton{XPolynomialRingXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{(p +q)^2 -p^2 -q^2 - 2*p*q\free{p }\free{q }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch8}
\begin{paste}{XPolynomialRingXmpPageFull8}{XPolynomialRingXmpPageEmpty8}
\pastebutton{XPolynomialRingXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{M := SquareMatrix(2,Fraction Integer)\bound{M }}
\indentrel{3}\begin{verbatim}
   (8)  SquareMatrix(2,Fraction(Integer))
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty8}
\begin{paste}{XPolynomialRingXmpPageEmpty8}{XPolynomialRingXmpPagePatch8}
\pastebutton{XPolynomialRingXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{M := SquareMatrix(2,Fraction Integer)\bound{M }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch9}
\begin{paste}{XPolynomialRingXmpPageFull9}{XPolynomialRingXmpPageEmpty9}
\pastebutton{XPolynomialRingXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{poly1:= XPR(M,Word)\free{Word }\free{M }\bound{poly1 }}
\indentrel{3}\begin{verbatim}
   (9)
  XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeM
  onoid(Symbol))
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty9}
\begin{paste}{XPolynomialRingXmpPageEmpty9}{XPolynomialRingXmpPagePatch9}
\pastebutton{XPolynomialRingXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{poly1:= XPR(M,Word)\free{Word }\free{M }\bound{poly1 }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch10}
\begin{paste}{XPolynomialRingXmpPageFull10}{XPolynomialRingXmpPageEmpty10}
\pastebutton{XPolynomialRingXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{m1:M := matrix [[i*j^2 for i in 1..2] for j in 1..2]\free{M }\bound{m1 }}
\indentrel{3}\begin{verbatim}
         +1  2+
   (10)  |    |
         +4  8+
                Type: SquareMatrix(2,Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty10}
\begin{paste}{XPolynomialRingXmpPageEmpty10}{XPolynomialRingXmpPagePatch10}
\pastebutton{XPolynomialRingXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{m1:M := matrix [[i*j^2 for i in 1..2] for j in 1..2]\free{M }\bound{m1 }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch11}
\begin{paste}{XPolynomialRingXmpPageFull11}{XPolynomialRingXmpPageEmpty11}
\pastebutton{XPolynomialRingXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{m2:M := m1 - 5/4\free{M }\free{m1 }\bound{m2 }}
\indentrel{3}\begin{verbatim}
         +  1    +
         |- -  2 |
         |  4    |
   (11)  |       |
         |     27|
         | 4   --|
         +      4+
                Type: SquareMatrix(2,Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty11}
\begin{paste}{XPolynomialRingXmpPageEmpty11}{XPolynomialRingXmpPagePatch11}
\pastebutton{XPolynomialRingXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{m2:M := m1 - 5/4\free{M }\free{m1 }\bound{m2 }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch12}
\begin{paste}{XPolynomialRingXmpPageFull12}{XPolynomialRingXmpPageEmpty12}
\pastebutton{XPolynomialRingXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{m3: M := m2^2\free{M }\free{m2 }\bound{m3 }}
\indentrel{3}\begin{verbatim}
         +129     +
         |---  13 |
         | 16     |
   (12)  |        |
         |     857|
         |26   ---|
         +      16+
                Type: SquareMatrix(2,Fraction(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty12}
\begin{paste}{XPolynomialRingXmpPageEmpty12}{XPolynomialRingXmpPagePatch12}
\pastebutton{XPolynomialRingXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{m3: M := m2^2\free{M }\free{m2 }\bound{m3 }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch13}
\begin{paste}{XPolynomialRingXmpPageFull13}{XPolynomialRingXmpPageEmpty13}
\pastebutton{XPolynomialRingXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{pm:poly1 := m1*x + m2*y + m3*z - 2/3\free{poly1 }\free{m1 }\free{m2 }\free{m3 }\bound{pm }}
\indentrel{3}\begin{verbatim}
         +  2     +             +  1    +    +129     +
         |- -   0 |             |- -  2 |    |---  13 |
         |  3     |   +1  2+    |  4    |    | 16     |
   (13)  |        | + |    |x + |       |y + |        |z
         |       2|   +4  8+    |     27|    |     857|
         | 0   - -|             | 4   --|    |26   ---|
         +       3+             +      4+    +      16+
Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty13}
\begin{paste}{XPolynomialRingXmpPageEmpty13}{XPolynomialRingXmpPagePatch13}
\pastebutton{XPolynomialRingXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{pm:poly1 := m1*x + m2*y + m3*z - 2/3\free{poly1 }\free{m1 }\free{m2 }\free{m3 }\bound{pm }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch14}
\begin{paste}{XPolynomialRingXmpPageFull14}{XPolynomialRingXmpPageEmpty14}
\pastebutton{XPolynomialRingXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{qm:poly1 := pm - m1*x\free{m1 }\free{pm }\bound{qm }}
\indentrel{3}\begin{verbatim}
         +  2     +   +  1    +    +129     +
         |- -   0 |   |- -  2 |    |---  13 |
         |  3     |   |  4    |    | 16     |
   (14)  |        | + |       |y + |        |z
         |       2|   |     27|    |     857|
         | 0   - -|   | 4   --|    |26   ---|
         +       3+   +      4+    +      16+
Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty14}
\begin{paste}{XPolynomialRingXmpPageEmpty14}{XPolynomialRingXmpPagePatch14}
\pastebutton{XPolynomialRingXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{qm:poly1 := pm - m1*x\free{m1 }\free{pm }\bound{qm }}
\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPagePatch15}
\begin{paste}{XPolynomialRingXmpPageFull15}{XPolynomialRingXmpPageEmpty15}
\pastebutton{XPolynomialRingXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{qm^3\bound{qm }}
\indentrel{3}\begin{verbatim}
   (15)
     +   8      +   +  1  8+    +43   52 +
     |- --   0  |   |- -  -|    |--   -- |
     |  27      |   |  3  3|    | 4    3 |
     |          | + |      |y + |        |z
     |         8|   |16    |    |104  857|
     | 0    - --|   |--   9|    |---  ---|
     +        27+   + 3    +    + 3    12+
   + 
     +  129       +     +  3199     831 +
     |- ---  - 26 |     |- ----   - --- |
     |   8        | 2   |   32       4  |
     |            |y  + |               |y z
     |         857|     |  831     26467|
     |- 52   - ---|     |- ---   - -----|
     +          8 +     +   2        32 +
   + 
     +  3199     831 +      +  103169     6409 +
     |- ----   - --- |      |- ------   - ---- |
     |   32       4  |      |    128        4  | 2
     |               |z y + |                  |z
     |  831     26467|      |   6409     820977|
     |- ---   - -----|      | - ----   - ------|
     +   2        32 +      +     2        128 +
   + 
     +3199   831 +     +103169   6409 +
     |----   --- |     |------   ---- |
     | 64     8  | 3   |  256      8  | 2
     |           |y  + |              |y z
     |831   26467|     | 6409   820977|
     |---   -----|     | ----   ------|
     + 4      64 +     +   4      256 +
   + 
     +103169   6409 +        +3178239   795341 +
     |------   ---- |        |-------   ------ |
     |  256      8  |        |  1024      128  |   2
     |              |y z y + |                 |y z
     | 6409   820977|        |795341   25447787|
     | ----   ------|        |------   --------|
     +   4      256 +        +  64       1024  +
   + 
     +103169   6409 +       +3178239   795341 +
     |------   ---- |       |-------   ------ |
     |  256      8  |   2   |  1024      128  |
     |              |z y  + |                 |z y z
     | 6409   820977|       |795341   25447787|
     | ----   ------|       |------   --------|
     +   4      256 +       +  64       1024  +
   + 
     +3178239   795341 +      +98625409  12326223 +
     |-------   ------ |      |--------  -------- |
     |  1024      128  | 2    |  4096       256   | 3
     |                 |z y + |                   |z
     |795341   25447787|      |12326223  788893897|
     |------   --------|      |--------  ---------|
     +  64       1024  +      +   128       4096  +
Type: XPolynomialRing(SquareMatrix(2,Fraction(Integer)),FreeMonoid(Symbol))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{XPolynomialRingXmpPageEmpty15}
\begin{paste}{XPolynomialRingXmpPageEmpty15}{XPolynomialRingXmpPagePatch15}
\pastebutton{XPolynomialRingXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{qm^3\bound{qm }}
\end{paste}\end{patch}

