\begin{patch}{LexTriangularPackageXmpPagePatch1}
\begin{paste}{LexTriangularPackageXmpPageFull1}{LexTriangularPackageXmpPageEmpty1}
\pastebutton{LexTriangularPackageXmpPageFull1}{\hidepaste}
\tab{5}\spadcommand{R := Integer\bound{R }}
\indentrel{3}\begin{verbatim}
   (1)  Integer
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty1}
\begin{paste}{LexTriangularPackageXmpPageEmpty1}{LexTriangularPackageXmpPagePatch1}
\pastebutton{LexTriangularPackageXmpPageEmpty1}{\showpaste}
\tab{5}\spadcommand{R := Integer\bound{R }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch2}
\begin{paste}{LexTriangularPackageXmpPageFull2}{LexTriangularPackageXmpPageEmpty2}
\pastebutton{LexTriangularPackageXmpPageFull2}{\hidepaste}
\tab{5}\spadcommand{ls : List Symbol := [a,b,c,d,e,f]\bound{ls }}
\indentrel{3}\begin{verbatim}
   (2)  [a, b, c, d, e, f]
                                     Type: List(Symbol)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty2}
\begin{paste}{LexTriangularPackageXmpPageEmpty2}{LexTriangularPackageXmpPagePatch2}
\pastebutton{LexTriangularPackageXmpPageEmpty2}{\showpaste}
\tab{5}\spadcommand{ls : List Symbol := [a,b,c,d,e,f]\bound{ls }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch3}
\begin{paste}{LexTriangularPackageXmpPageFull3}{LexTriangularPackageXmpPageEmpty3}
\pastebutton{LexTriangularPackageXmpPageFull3}{\hidepaste}
\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
\indentrel{3}\begin{verbatim}
   (3)  OrderedVariableList([a, b, c, d, e, f])
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty3}
\begin{paste}{LexTriangularPackageXmpPageEmpty3}{LexTriangularPackageXmpPagePatch3}
\pastebutton{LexTriangularPackageXmpPageEmpty3}{\showpaste}
\tab{5}\spadcommand{V := OVAR(ls)\free{ls }\bound{V }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch4}
\begin{paste}{LexTriangularPackageXmpPageFull4}{LexTriangularPackageXmpPageEmpty4}
\pastebutton{LexTriangularPackageXmpPageFull4}{\hidepaste}
\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
\indentrel{3}\begin{verbatim}
   (4)
   NewSparseMultivariatePolynomial
      Integer
  ,
      OrderedVariableList([a, b, c, d, e, f])
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty4}
\begin{paste}{LexTriangularPackageXmpPageEmpty4}{LexTriangularPackageXmpPagePatch4}
\pastebutton{LexTriangularPackageXmpPageEmpty4}{\showpaste}
\tab{5}\spadcommand{P := NSMP(R, V)\free{R }\free{V }\bound{P }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch5}
\begin{paste}{LexTriangularPackageXmpPageFull5}{LexTriangularPackageXmpPageEmpty5}
\pastebutton{LexTriangularPackageXmpPageFull5}{\hidepaste}
\tab{5}\spadcommand{p1: P := a*b*c*d*e*f - 1\free{P }\bound{p1 }}
\indentrel{3}\begin{verbatim}
   (5)  f e d c b a - 1
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty5}
\begin{paste}{LexTriangularPackageXmpPageEmpty5}{LexTriangularPackageXmpPagePatch5}
\pastebutton{LexTriangularPackageXmpPageEmpty5}{\showpaste}
\tab{5}\spadcommand{p1: P := a*b*c*d*e*f - 1\free{P }\bound{p1 }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch6}
\begin{paste}{LexTriangularPackageXmpPageFull6}{LexTriangularPackageXmpPageEmpty6}
\pastebutton{LexTriangularPackageXmpPageFull6}{\hidepaste}
\tab{5}\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f\free{P }\bound{p2 }}
\indentrel{3}\begin{verbatim}
   (6)
   ((((e + f)d + f e)c + f e d)b + f e d c)a + f e d c b
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty6}
\begin{paste}{LexTriangularPackageXmpPageEmpty6}{LexTriangularPackageXmpPagePatch6}
\pastebutton{LexTriangularPackageXmpPageEmpty6}{\showpaste}
\tab{5}\spadcommand{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f\free{P }\bound{p2 }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch7}
\begin{paste}{LexTriangularPackageXmpPageFull7}{LexTriangularPackageXmpPageEmpty7}
\pastebutton{LexTriangularPackageXmpPageFull7}{\hidepaste}
\tab{5}\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f\free{P }\bound{p3 }}
\indentrel{3}\begin{verbatim}
   (7)  (((d + f)c + f e)b + f e d)a + e d c b + f e d c
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty7}
\begin{paste}{LexTriangularPackageXmpPageEmpty7}{LexTriangularPackageXmpPagePatch7}
\pastebutton{LexTriangularPackageXmpPageEmpty7}{\showpaste}
\tab{5}\spadcommand{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f\free{P }\bound{p3 }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch8}
\begin{paste}{LexTriangularPackageXmpPageFull8}{LexTriangularPackageXmpPageEmpty8}
\pastebutton{LexTriangularPackageXmpPageFull8}{\hidepaste}
\tab{5}\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f\free{P }\bound{p4 }}
\indentrel{3}\begin{verbatim}
   (8)  ((c + f)b + f e)a + d c b + e d c + f e d
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty8}
\begin{paste}{LexTriangularPackageXmpPageEmpty8}{LexTriangularPackageXmpPagePatch8}
\pastebutton{LexTriangularPackageXmpPageEmpty8}{\showpaste}
\tab{5}\spadcommand{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f\free{P }\bound{p4 }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch9}
\begin{paste}{LexTriangularPackageXmpPageFull9}{LexTriangularPackageXmpPageEmpty9}
\pastebutton{LexTriangularPackageXmpPageFull9}{\hidepaste}
\tab{5}\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f\free{P }\bound{p5 }}
\indentrel{3}\begin{verbatim}
   (9)  (b + f)a + c b + d c + e d + f e
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty9}
\begin{paste}{LexTriangularPackageXmpPageEmpty9}{LexTriangularPackageXmpPagePatch9}
\pastebutton{LexTriangularPackageXmpPageEmpty9}{\showpaste}
\tab{5}\spadcommand{p5: P := a*b + a*f + b*c + c*d + d*e + e*f\free{P }\bound{p5 }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch10}
\begin{paste}{LexTriangularPackageXmpPageFull10}{LexTriangularPackageXmpPageEmpty10}
\pastebutton{LexTriangularPackageXmpPageFull10}{\hidepaste}
\tab{5}\spadcommand{p6: P := a + b + c + d + e + f\free{P }\bound{p6 }}
\indentrel{3}\begin{verbatim}
   (10)  a + b + c + d + e + f
Type: NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty10}
\begin{paste}{LexTriangularPackageXmpPageEmpty10}{LexTriangularPackageXmpPagePatch10}
\pastebutton{LexTriangularPackageXmpPageEmpty10}{\showpaste}
\tab{5}\spadcommand{p6: P := a + b + c + d + e + f\free{P }\bound{p6 }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch11}
\begin{paste}{LexTriangularPackageXmpPageFull11}{LexTriangularPackageXmpPageEmpty11}
\pastebutton{LexTriangularPackageXmpPageFull11}{\hidepaste}
\tab{5}\spadcommand{lp := [p1, p2, p3, p4, p5, p6]\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\bound{lp }}
\indentrel{3}\begin{verbatim}
   (11)
   [f e d c b a - 1,
    ((((e + f)d + f e)c + f e d)b + f e d c)a + f e d c b,
    (((d + f)c + f e)b + f e d)a + e d c b + f e d c,
    ((c + f)b + f e)a + d c b + e d c + f e d,
    (b + f)a + c b + d c + e d + f e,
    a + b + c + d + e + f]
Type: List(NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f])))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty11}
\begin{paste}{LexTriangularPackageXmpPageEmpty11}{LexTriangularPackageXmpPagePatch11}
\pastebutton{LexTriangularPackageXmpPageEmpty11}{\showpaste}
\tab{5}\spadcommand{lp := [p1, p2, p3, p4, p5, p6]\free{p1 }\free{p2 }\free{p3 }\free{p4 }\free{p5 }\free{p6 }\bound{lp }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch12}
\begin{paste}{LexTriangularPackageXmpPageFull12}{LexTriangularPackageXmpPageEmpty12}
\pastebutton{LexTriangularPackageXmpPageFull12}{\hidepaste}
\tab{5}\spadcommand{lextripack := LEXTRIPK(R,ls)\free{R }\free{ls }\bound{lextripack }}
\indentrel{3}\begin{verbatim}
   (12)
   LexTriangularPackage(Integer,[a, b, c, d, e, f])
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty12}
\begin{paste}{LexTriangularPackageXmpPageEmpty12}{LexTriangularPackageXmpPagePatch12}
\pastebutton{LexTriangularPackageXmpPageEmpty12}{\showpaste}
\tab{5}\spadcommand{lextripack := LEXTRIPK(R,ls)\free{R }\free{ls }\bound{lextripack }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch13}
\begin{paste}{LexTriangularPackageXmpPageFull13}{LexTriangularPackageXmpPageEmpty13}
\pastebutton{LexTriangularPackageXmpPageFull13}{\hidepaste}
\tab{5}\spadcommand{lg := groebner(lp)$lextripack;\free{lp }\free{lextripack }\bound{lg }}
\indentrel{3}\begin{verbatim}
Type: List(NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f])))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty13}
\begin{paste}{LexTriangularPackageXmpPageEmpty13}{LexTriangularPackageXmpPagePatch13}
\pastebutton{LexTriangularPackageXmpPageEmpty13}{\showpaste}
\tab{5}\spadcommand{lg := groebner(lp)$lextripack;\free{lp }\free{lextripack }\bound{lg }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch14}
\begin{paste}{LexTriangularPackageXmpPageFull14}{LexTriangularPackageXmpPageEmpty14}
\pastebutton{LexTriangularPackageXmpPageFull14}{\hidepaste}
\tab{5}\spadcommand{lexTriangular(lg,false)$lextripack\free{lg }\free{lextripack }}
\indentrel{3}\begin{verbatim}
   (14)
   [
       6
     {f  + 1,
       6        5      2 4      3 3      4 2      5
      e  - 3 f e  + 3 f e  - 4 f e  + 3 f e  - 3 f e - 1,
             2 5      3 4      4 3      5 2
      3 d + f e  - 4 f e  + 4 f e  - 2 f e  - 2 e + 2 f,
      c + f,
               2 5      3 4      4 3       5 2
      3 b + 2 f e  - 5 f e  + 5 f e  - 10 f e  - 4 e + 7 f
       ,
           2 5      3 4      4 3      5 2
      a - f e  + 3 f e  - 3 f e  + 4 f e  + 3 e - 3 f}
     ,

       6                     2            2
     {f  - 1, e - f, d - f, c  + 4 f c + f ,
                          2
      (c - f)b - f c - 5 f , a + b + c + 3 f}
     ,

       6                            2            2
     {f  - 1, e - f, d - f, c - f, b  + 4 f b + f ,
      a + b + 4 f}
     ,

       6              2            2
     {f  - 1, e - f, d  + 4 f d + f ,
                          2
      (d - f)c - f d - 5 f , b - f, a + c + d + 3 f}
     ,

     {
          36         30           24           18
         f   - 2554 f   - 399709 f   - 502276 f
       + 
                   12         6
         - 399709 f   - 2554 f  + 1
       ,

                     12              2
         (161718564 f   - 161718564)e
       + 
                       31               25
             - 504205 f   + 1287737951 f
           + 
                           19                 13
             201539391380 f   + 253982817368 f
           + 
                           7
             201940704665 f  + 1574134601 f
        *
           e
       + 
                    32               26
         - 2818405 f   + 7198203911 f
       + 
                        20                  14
         1126548149060 f   + 1416530563364 f
       + 
                        8               2
         1127377589345 f  + 7988820725 f
       ,

                        6
         (693772639560 f  - 693772639560)d
       + 
                         2 5                  3 4
         - 462515093040 f e  + 1850060372160 f e
       + 
                          4 3
         - 1850060372160 f e
       + 
                            11                   5  2
         (- 24513299931120 f   - 23588269745040 f )e
       + 
                          30                  24
             - 890810428 f   + 2275181044754 f
           + 
                              18                    12
             355937263869776 f   + 413736880104344 f
           + 
                              6
             342849304487996 f  + 3704966481878
        *
           e
       + 
                       31                   25
         - 4163798003 f   + 10634395752169 f
       + 
                           19                     13
         1664161760192806 f   + 2079424391370694 f
       + 
                           7
         1668153650635921 f  + 10924274392693 f
       ,

                       6                            31
         (12614047992 f  - 12614047992)c - 7246825 f
       + 
                      25                  19
         18508536599 f   + 2896249516034 f
       + 
                        13                  7
         3581539649666 f   + 2796477571739 f
       + 
         - 48094301893 f
       ,

                        6
         (693772639560 f  - 693772639560)b
       + 
                         2 5                  3 4
         - 925030186080 f e  + 2312575465200 f e
       + 
                          4 3
         - 2312575465200 f e
       + 
                            11                   5  2
         (- 40007555547960 f   - 35382404617560 f )e
       + 
                           30                  24
             - 3781280823 f   + 9657492291789 f
           + 
                               18                     12
             1511158913397906 f   + 1837290892286154 f
           + 
                               6
             1487216006594361 f  + 8077238712093
        *
           e
       + 
                       31                   25
         - 9736390478 f   + 24866827916734 f
       + 
                           19                     13
         3891495681905296 f   + 4872556418871424 f
       + 
                           7
         3904047887269606 f  + 27890075838538 f
       ,
      a + b + c + d + e + f}
     ,

       6       2            2                      2
     {f  - 1, e  + 4 f e + f , (e - f)d - f e - 5 f ,
      c - f, b - f, a + d + e + 3 f}
     ]
        Type: List(RegularChain(Integer,[a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty14}
\begin{paste}{LexTriangularPackageXmpPageEmpty14}{LexTriangularPackageXmpPagePatch14}
\pastebutton{LexTriangularPackageXmpPageEmpty14}{\showpaste}
\tab{5}\spadcommand{lexTriangular(lg,false)$lextripack\free{lg }\free{lextripack }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch15}
\begin{paste}{LexTriangularPackageXmpPageFull15}{LexTriangularPackageXmpPageEmpty15}
\pastebutton{LexTriangularPackageXmpPageFull15}{\hidepaste}
\tab{5}\spadcommand{lts := lexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }\bound{lts }}
\indentrel{3}\begin{verbatim}
   (15)
   [
       6
     {f  + 1,
       6        5      2 4      3 3      4 2      5
      e  - 3 f e  + 3 f e  - 4 f e  + 3 f e  - 3 f e - 1,
             2 5      3 4      4 3      5 2
      3 d + f e  - 4 f e  + 4 f e  - 2 f e  - 2 e + 2 f,
      c + f,
               2 5      3 4      4 3       5 2
      3 b + 2 f e  - 5 f e  + 5 f e  - 10 f e  - 4 e + 7 f
       ,
           2 5      3 4      4 3      5 2
      a - f e  + 3 f e  - 3 f e  + 4 f e  + 3 e - 3 f}
     ,

       6                     2            2
     {f  - 1, e - f, d - f, c  + 4 f c + f , b + c + 4 f,
      a - f}
     ,

       6                            2            2
     {f  - 1, e - f, d - f, c - f, b  + 4 f b + f ,
      a + b + 4 f}
     ,

       6              2            2
     {f  - 1, e - f, d  + 4 f d + f , c + d + 4 f, b - f,
      a - f}
     ,

     {
          36         30           24           18
         f   - 2554 f   - 399709 f   - 502276 f
       + 
                   12         6
         - 399709 f   - 2554 f  + 1
       ,

                        2
         1387545279120 e
       + 
                         31                   25
             4321823003 f   - 11037922310209 f
           + 
                                 19                     13
             - 1727506390124986 f   - 2176188913464634 f
           + 
                                 7
             - 1732620732685741 f  - 13506088516033 f
        *
           e
       + 
                      32                   26
         24177661775 f   - 61749727185325 f
       + 
                             20                      14
         - 9664082618092450 f   - 12152237485813570 f
       + 
                             8                   2
         - 9672870290826025 f  - 68544102808525 f
       ,

         1387545279120 d
       + 
                           30                  24
             - 1128983050 f   + 2883434331830 f
           + 
                              18                    12
             451234998755840 f   + 562426491685760 f
           + 
                              6
             447129055314890 f  - 165557857270
        *
           e
       + 
                       31                  25
         - 1816935351 f   + 4640452214013 f
       + 
                          19                    13
         726247129626942 f   + 912871801716798 f
       + 
                          7
         726583262666877 f  + 4909358645961 f
       ,

                                      31
         1387545279120 c + 778171189 f
       + 
                          25                    19
         - 1987468196267 f   - 310993556954378 f
       + 
                            13                    7
         - 383262822316802 f   - 300335488637543 f
       + 
         5289595037041 f
       ,

         1387545279120 b
       + 
                         30                  24
             1128983050 f   - 2883434331830 f
           + 
                                18                    12
             - 451234998755840 f   - 562426491685760 f
           + 
                                6
             - 447129055314890 f  + 165557857270
        *
           e
       + 
                       31                  25
         - 3283058841 f   + 8384938292463 f
       + 
                           19                     13
         1312252817452422 f   + 1646579934064638 f
       + 
                           7
         1306372958656407 f  + 4694680112151 f
       ,

         1387545279120 a + 1387545279120 e
       + 
                     31                   25
         4321823003 f   - 11037922310209 f
       + 
                             19                     13
         - 1727506390124986 f   - 2176188913464634 f
       + 
                             7
         - 1732620732685741 f  - 13506088516033 f
       }
     ,

       6       2            2
     {f  - 1, e  + 4 f e + f , d + e + 4 f, c - f, b - f,
      a - f}
     ]
        Type: List(RegularChain(Integer,[a,b,c,d,e,f]))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty15}
\begin{paste}{LexTriangularPackageXmpPageEmpty15}{LexTriangularPackageXmpPagePatch15}
\pastebutton{LexTriangularPackageXmpPageEmpty15}{\showpaste}
\tab{5}\spadcommand{lts := lexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }\bound{lts }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch16}
\begin{paste}{LexTriangularPackageXmpPageFull16}{LexTriangularPackageXmpPageEmpty16}
\pastebutton{LexTriangularPackageXmpPageFull16}{\hidepaste}
\tab{5}\spadcommand{[[init(p) for p in (ts :: List(P))] for ts in lts]\free{lts }}
\indentrel{3}\begin{verbatim}
   (16)
   [[1, 3, 1, 3, 1, 1], [1, 1, 1, 1, 1, 1],
    [1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1],

     [1387545279120, 1387545279120, 1387545279120,
      1387545279120, 1387545279120, 1]
     ,
    [1, 1, 1, 1, 1, 1]]
Type: List(List(NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty16}
\begin{paste}{LexTriangularPackageXmpPageEmpty16}{LexTriangularPackageXmpPagePatch16}
\pastebutton{LexTriangularPackageXmpPageEmpty16}{\showpaste}
\tab{5}\spadcommand{[[init(p) for p in (ts :: List(P))] for ts in lts]\free{lts }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch17}
\begin{paste}{LexTriangularPackageXmpPageFull17}{LexTriangularPackageXmpPageEmpty17}
\pastebutton{LexTriangularPackageXmpPageFull17}{\hidepaste}
\tab{5}\spadcommand{squareFreeLexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }}
\indentrel{3}\begin{verbatim}
   (17)
   [
       6
     {f  + 1,
       6        5      2 4      3 3      4 2      5
      e  - 3 f e  + 3 f e  - 4 f e  + 3 f e  - 3 f e - 1,
             2 5      3 4      4 3      5 2
      3 d + f e  - 4 f e  + 4 f e  - 2 f e  - 2 e + 2 f,
      c + f,
               2 5      3 4      4 3       5 2
      3 b + 2 f e  - 5 f e  + 5 f e  - 10 f e  - 4 e + 7 f
       ,
           2 5      3 4      4 3      5 2
      a - f e  + 3 f e  - 3 f e  + 4 f e  + 3 e - 3 f}
     ,

       6                     2            2
     {f  - 1, e - f, d - f, c  + 4 f c + f , b + c + 4 f,
      a - f}
     ,

       6                            2            2
     {f  - 1, e - f, d - f, c - f, b  + 4 f b + f ,
      a + b + 4 f}
     ,

       6              2            2
     {f  - 1, e - f, d  + 4 f d + f , c + d + 4 f, b - f,
      a - f}
     ,

     {
          36         30           24           18
         f   - 2554 f   - 399709 f   - 502276 f
       + 
                   12         6
         - 399709 f   - 2554 f  + 1
       ,

                        2
         1387545279120 e
       + 
                         31                   25
             4321823003 f   - 11037922310209 f
           + 
                                 19                     13
             - 1727506390124986 f   - 2176188913464634 f
           + 
                                 7
             - 1732620732685741 f  - 13506088516033 f
        *
           e
       + 
                      32                   26
         24177661775 f   - 61749727185325 f
       + 
                             20                      14
         - 9664082618092450 f   - 12152237485813570 f
       + 
                             8                   2
         - 9672870290826025 f  - 68544102808525 f
       ,

         1387545279120 d
       + 
                           30                  24
             - 1128983050 f   + 2883434331830 f
           + 
                              18                    12
             451234998755840 f   + 562426491685760 f
           + 
                              6
             447129055314890 f  - 165557857270
        *
           e
       + 
                       31                  25
         - 1816935351 f   + 4640452214013 f
       + 
                          19                    13
         726247129626942 f   + 912871801716798 f
       + 
                          7
         726583262666877 f  + 4909358645961 f
       ,

                                      31
         1387545279120 c + 778171189 f
       + 
                          25                    19
         - 1987468196267 f   - 310993556954378 f
       + 
                            13                    7
         - 383262822316802 f   - 300335488637543 f
       + 
         5289595037041 f
       ,

         1387545279120 b
       + 
                         30                  24
             1128983050 f   - 2883434331830 f
           + 
                                18                    12
             - 451234998755840 f   - 562426491685760 f
           + 
                                6
             - 447129055314890 f  + 165557857270
        *
           e
       + 
                       31                  25
         - 3283058841 f   + 8384938292463 f
       + 
                           19                     13
         1312252817452422 f   + 1646579934064638 f
       + 
                           7
         1306372958656407 f  + 4694680112151 f
       ,

         1387545279120 a + 1387545279120 e
       + 
                     31                   25
         4321823003 f   - 11037922310209 f
       + 
                             19                     13
         - 1727506390124986 f   - 2176188913464634 f
       + 
                             7
         - 1732620732685741 f  - 13506088516033 f
       }
     ,

       6       2            2
     {f  - 1, e  + 4 f e + f , d + e + 4 f, c - f, b - f,
      a - f}
     ]
Type: List(SquareFreeRegularTriangularSet(Integer,IndexedExponents(OrderedVariableList([a,b,c,d,e,f])),OrderedVariableList([a,b,c,d,e,f]),NewSparseMultivariatePolynomial(Integer,OrderedVariableList([a,b,c,d,e,f]))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty17}
\begin{paste}{LexTriangularPackageXmpPageEmpty17}{LexTriangularPackageXmpPagePatch17}
\pastebutton{LexTriangularPackageXmpPageEmpty17}{\showpaste}
\tab{5}\spadcommand{squareFreeLexTriangular(lg,true)$lextripack\free{lg }\free{lextripack }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch18}
\begin{paste}{LexTriangularPackageXmpPageFull18}{LexTriangularPackageXmpPageEmpty18}
\pastebutton{LexTriangularPackageXmpPageFull18}{\hidepaste}
\tab{5}\spadcommand{reduce(+,[degree(ts) for ts in lts])\free{lts }}
\indentrel{3}\begin{verbatim}
   (18)  156
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty18}
\begin{paste}{LexTriangularPackageXmpPageEmpty18}{LexTriangularPackageXmpPagePatch18}
\pastebutton{LexTriangularPackageXmpPageEmpty18}{\showpaste}
\tab{5}\spadcommand{reduce(+,[degree(ts) for ts in lts])\free{lts }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch19}
\begin{paste}{LexTriangularPackageXmpPageFull19}{LexTriangularPackageXmpPageEmpty19}
\pastebutton{LexTriangularPackageXmpPageFull19}{\hidepaste}
\tab{5}\spadcommand{ls2 : List Symbol := concat(ls,new()$Symbol)\free{ls }\bound{ls2 }}
\indentrel{3}\begin{verbatim}
   (19)  [a, b, c, d, e, f, %A]
                                     Type: List(Symbol)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty19}
\begin{paste}{LexTriangularPackageXmpPageEmpty19}{LexTriangularPackageXmpPagePatch19}
\pastebutton{LexTriangularPackageXmpPageEmpty19}{\showpaste}
\tab{5}\spadcommand{ls2 : List Symbol := concat(ls,new()$Symbol)\free{ls }\bound{ls2 }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch20}
\begin{paste}{LexTriangularPackageXmpPageFull20}{LexTriangularPackageXmpPageEmpty20}
\pastebutton{LexTriangularPackageXmpPageFull20}{\hidepaste}
\tab{5}\spadcommand{zdpack := ZDSOLVE(R,ls,ls2)\free{R }\free{ls }\free{ls2 }\bound{zdpack }}
\indentrel{3}\begin{verbatim}
   (20)
   ZeroDimensionalSolvePackage
      Integer
  ,
      [a, b, c, d, e, f]
  ,
      [a, b, c, d, e, f, %A]
                                             Type: Type
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty20}
\begin{paste}{LexTriangularPackageXmpPageEmpty20}{LexTriangularPackageXmpPagePatch20}
\pastebutton{LexTriangularPackageXmpPageEmpty20}{\showpaste}
\tab{5}\spadcommand{zdpack := ZDSOLVE(R,ls,ls2)\free{R }\free{ls }\free{ls2 }\bound{zdpack }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch21}
\begin{paste}{LexTriangularPackageXmpPageFull21}{LexTriangularPackageXmpPageEmpty21}
\pastebutton{LexTriangularPackageXmpPageFull21}{\hidepaste}
\tab{5}\spadcommand{concat [univariateSolve(ts)$zdpack for ts in lts];\free{lts }\free{zdpack }}
\indentrel{3}\begin{verbatim}
Type: List(Record(complexRoots: SparseUnivariatePolynomial(Integer),coordinates: List(Polynomial(Integer))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty21}
\begin{paste}{LexTriangularPackageXmpPageEmpty21}{LexTriangularPackageXmpPagePatch21}
\pastebutton{LexTriangularPackageXmpPageEmpty21}{\showpaste}
\tab{5}\spadcommand{concat [univariateSolve(ts)$zdpack for ts in lts];\free{lts }\free{zdpack }}
\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPagePatch22}
\begin{paste}{LexTriangularPackageXmpPageFull22}{LexTriangularPackageXmpPageEmpty22}
\pastebutton{LexTriangularPackageXmpPageFull22}{\hidepaste}
\tab{5}\spadcommand{concat [realSolve(ts)$zdpack for ts in lts];\free{lts }\free{zdpack }}
\indentrel{3}\begin{verbatim}
       Type: List(List(RealClosure(Fraction(Integer))))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{LexTriangularPackageXmpPageEmpty22}
\begin{paste}{LexTriangularPackageXmpPageEmpty22}{LexTriangularPackageXmpPagePatch22}
\pastebutton{LexTriangularPackageXmpPageEmpty22}{\showpaste}
\tab{5}\spadcommand{concat [realSolve(ts)$zdpack for ts in lts];\free{lts }\free{zdpack }}
\end{paste}\end{patch}

