\begin{chapter}{Direkter Beweis des Satzes von Morley}\label{Direkter Beweis Satz von Morley}
In diesem Anhang ist die Methode beschrieben mit der ich zuerst versucht den Satz von Morley zu beweisen, leider war
\ma\ nicht in der Lage die \gb\ zu berechnen. Deshalb habe ich die Konstruktion des zweiten Morleydreiecks benutzt, um
den Satz zu beweisen. Wie sich am Ende herausstellte war Reduce in der Lage diesen Beweis durchzuf{\"u}hren, um dies
nachvollziehen zu k{\"o}nnen findet sich am Ende des Anhangs der Reduce-Quelltext.

\dispTFinmath{ \MathBegin{MathArray}{l}
\Mvariable{vars}:=\{\Mvariable{x1},\Mvariable{x2},\Mvariable{x3},\Mvariable{x4},\Mvariable{x5},\Mvariable{x6},y\};  \\
\noalign{\vspace{0.625ex}}  \\
\Mvariable{parameters}:=\{\Mvariable{u1},\Mvariable{u2},\Mvariable{u3}\}; \MathEnd{MathArray} }

\dispTFinmath{ \MathBegin{MathArray}{l}
\Mfunction{Unprotect}[C,D,E];  \\
A=\Muserfunction{point}[0,0];\\
B=\Muserfunction{point}[\Mvariable{u1},0];  \\
C=\Muserfunction{point}[\Mvariable{u2},\Mvariable{u3}];  \\
D=\Muserfunction{point}[\Mvariable{x5},\Mvariable{x6}];\\
E=\Muserfunction{point}[\Mvariable{x1},\Mvariable{x2}];  \\
F=\Muserfunction{point}[\Mvariable{x3},\Mvariable{x4}]; \MathEnd{MathArray} }

\dispTFinmath{ \MathBegin{MathArray}{l}
\Muserfunction{AddGleichung}[\Muserfunction{dreiteilung}[\Muserfunction{p3angle}[B,A,C],  \\
\noalign{\vspace{0.625ex}}
\hspace{2.em} \Muserfunction{p3angle}[B,A,D]]];\\
\MathEnd{MathArray} }

\dispTFinmath{ \MathBegin{MathArray}{l}
\Muserfunction{AddGleichung}[\Muserfunction{dreiteilung}[\Muserfunction{p3angle}[C,B,A],  \\
\noalign{\vspace{0.625ex}}
\hspace{2.em} \Muserfunction{p3angle}[C,B,e]]];\\
\MathEnd{MathArray} }

\dispTFinmath{ \MathBegin{MathArray}{l}
\Muserfunction{AddGleichung}[\Muserfunction{dreiteilung}[\Muserfunction{p3angle}[A,C,B],  \\
\noalign{\vspace{0.625ex}}
\hspace{2.em} \Muserfunction{p3angle}[A,C,F]]];\\
\MathEnd{MathArray} }

\pagebreak
\dispTFinmath{ \Mvariable{Bad}=\Muserfunction{sympoint}[B,\Muserfunction{ppline}[A,D]] }

\dispTFoutmath{ \Big\{\frac{\Mvariable{u1}\multsp {{\Mvariable{x5}}^2}-\Mvariable{u1}\multsp
{{\Mvariable{x6}}^2}}{{{\Mvariable{x5}}^2}+{{\Mvariable{x6}}^2}},\frac{2\multsp \Mvariable{u1}\multsp
\Mvariable{x5}\multsp \Mvariable{x6}}{{{\Mvariable{x5}}^2}+{{\Mvariable{x6}}^2}}\Big\} }

\dispTFinmath{ \Mvariable{Acf}=\Muserfunction{sympoint}[A,\Muserfunction{ppline}[C,F]] }

\dispTFoutmath{ \MathBegin{MathArray}{l} \Big\{\frac{2\multsp (\Mvariable{u3}-\Mvariable{x4})\multsp
(\Mvariable{u3}\multsp \Mvariable{x3}-\Mvariable{u2}\multsp \Mvariable{x4})}{{{\Mvariable{u2}}^2}-2\multsp
\Mvariable{x3}\multsp \Mvariable{u2}+{{\Mvariable{u3}}^2}+{{\Mvariable{x3}}^2}+{{\Mvariable{x4}}^2}-2\multsp
\Mvariable{u3}\multsp \Mvariable{x4}},
 \\
\noalign{\vspace{1.21875ex}} \hspace{1.em} \frac{2\multsp (\Mvariable{u2}-\Mvariable{x3})\multsp (\Mvariable{u2}\multsp
\Mvariable{x4}-\Mvariable{u3}\multsp \Mvariable{x3})}{{{\Mvariable{u2}}^2}-2\multsp
\Mvariable{x3}\multsp \Mvariable{u2}+{{\Mvariable{u3}}^2}+{{\Mvariable{x3}}^2}+{{\Mvariable{x4}}^2}-2\multsp \Mvariable{u3}\multsp \Mvariable{x4}}\Big\}\\
\MathEnd{MathArray} }

\dispTFinmath{ \Mvariable{Cbe}=\Muserfunction{sympoint}[C,\Muserfunction{ppline}[B,E]] }

\dispTFoutmath{ \MathBegin{MathArray}{l} \Big\{\frac{\Mvariable{u2}\multsp {{\Mvariable{u1}}^2}+2\multsp
{{\Mvariable{x2}}^2}\multsp \Mvariable{u1}-2\multsp \Mvariable{u2}\multsp \Mvariable{x1}\multsp \Mvariable{u1}-2\multsp
\Mvariable{u3}\multsp \Mvariable{x2}\multsp \Mvariable{u1}+\Mvariable{u2}\multsp
{{\Mvariable{x1}}^2}-\Mvariable{u2}\multsp {{\Mvariable{x2}}^2}+2\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp
\Mvariable{x2}}{{{\Mvariable{u1}}^2}-2\multsp \Mvariable{x1}\multsp
\Mvariable{u1}+{{\Mvariable{x1}}^2}+{{\Mvariable{x2}}^2}},
 \\
\noalign{\vspace{1.00781ex}} \hspace{1.em} (-\Mvariable{u3}\multsp {{\Mvariable{u1}}^2}+2\multsp \Mvariable{x2}\multsp
{{\Mvariable{u1}}^2}+2\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp \Mvariable{u1}-2\multsp \Mvariable{u2}\multsp
\Mvariable{x2}\multsp \Mvariable{u1}-2\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{u1}-
 \\
\noalign{\vspace{0.875ex}} \hspace{4.em} \Mvariable{u3}\multsp {{\Mvariable{x1}}^2}+\Mvariable{u3}\multsp
{{\Mvariable{x2}}^2}+2\multsp \Mvariable{u2}\multsp \Mvariable{x1}\multsp
\Mvariable{x2})/({{\Mvariable{u1}}^2}-2\multsp \Mvariable{x1}\multsp \Mvariable{u1}+{{\Mvariable{x1}}^2}+{{\Mvariable{x2}}^2})\Big\}\\
\MathEnd{MathArray} }

\dispTFinmath{
\Muserfunction{AddGleichung}[\Muserfunction{online}[E,\Muserfunction{ppline}[\Mvariable{Acf},C]]] }

\dispTFoutmath{ \MathBegin{MathArray}{l} -\Mvariable{x2}\multsp {{\Mvariable{u2}}^3}+2\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^3}+\Mvariable{u3}\multsp \Mvariable{x1}\multsp {{\Mvariable{u2}}^2}-2\multsp \Mvariable{u3}\multsp
\Mvariable{x3}\multsp {{\Mvariable{u2}}^2}+2\multsp \Mvariable{x2}\multsp \Mvariable{x3}\multsp
{{\Mvariable{u2}}^2}-2\multsp
\Mvariable{x1}\multsp \Mvariable{x4}\multsp {{\Mvariable{u2}}^2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} 2\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^2}+2\multsp \Mvariable{u3}\multsp {{\Mvariable{x3}}^2}\multsp \Mvariable{u2}-\Mvariable{x2}\multsp
{{\Mvariable{x3}}^2}\multsp \Mvariable{u2}-2\multsp \Mvariable{u3}\multsp {{\Mvariable{x4}}^2}\multsp
\Mvariable{u2}+\Mvariable{x2}\multsp
{{\Mvariable{x4}}^2}\multsp \Mvariable{u2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} {{\Mvariable{u3}}^2}\multsp \Mvariable{x2}\multsp \Mvariable{u2}+2\multsp
{{\Mvariable{u3}}^2}\multsp \Mvariable{x4}\multsp \Mvariable{u2}+2\multsp \Mvariable{x1}\multsp \Mvariable{x3}\multsp
\Mvariable{x4}\multsp \Mvariable{u2}-\Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{x3}}^2}+\Mvariable{u3}\multsp
\Mvariable{x1}\multsp {{\Mvariable{x4}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} {{\Mvariable{u3}}^3}\multsp \Mvariable{x1}-2\multsp
{{\Mvariable{u3}}^3}\multsp \Mvariable{x3}+2\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x2}\multsp
\Mvariable{x3}-2\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x1}\multsp \Mvariable{x4}+2\multsp
{{\Mvariable{u3}}^2}\multsp
\Mvariable{x3}\multsp \Mvariable{x4}-2\multsp \Mvariable{u3}\multsp \Mvariable{x2}\multsp \Mvariable{x3}\multsp \Mvariable{x4}\\
\MathEnd{MathArray} }

\dispTFinmath{ \Muserfunction{AddGleichung}[\Muserfunction{online}[F,\Muserfunction{ppline}[\Mvariable{Bad},A]]] }

\dispTFoutmath{ \Mvariable{x4}\multsp {{\Mvariable{x5}}^2}-2\multsp \Mvariable{x3}\multsp \Mvariable{x6}\multsp
\Mvariable{x5}-\Mvariable{x4}\multsp {{\Mvariable{x6}}^2} }

\dispTFinmath{ \Muserfunction{AddGleichung}[\Muserfunction{online}[D,\Muserfunction{ppline}[\Mvariable{Cbe},B]]] }

\dispTFoutmath{ \MathBegin{MathArray}{l} -\Mvariable{u3}\multsp {{\Mvariable{u1}}^3}+2\multsp \Mvariable{x2}\multsp
{{\Mvariable{u1}}^3}-\Mvariable{x6}\multsp {{\Mvariable{u1}}^3}+2\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{u1}}^2}-2\multsp \Mvariable{u2}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^2}-2\multsp
\Mvariable{x1}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} \Mvariable{u3}\multsp \Mvariable{x5}\multsp {{\Mvariable{u1}}^2}-2\multsp
\Mvariable{x2}\multsp \Mvariable{x5}\multsp {{\Mvariable{u1}}^2}+\Mvariable{u2}\multsp \Mvariable{x6}\multsp
{{\Mvariable{u1}}^2}+2\multsp \Mvariable{x1}\multsp \Mvariable{x6}\multsp {{\Mvariable{u1}}^2}-\Mvariable{u3}\multsp
{{\Mvariable{x1}}^2}\multsp
\Mvariable{u1}+  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} \Mvariable{u3}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{u1}+2\multsp
\Mvariable{u2}\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{u1}-2\multsp \Mvariable{u3}\multsp
\Mvariable{x1}\multsp \Mvariable{x5}\multsp \Mvariable{u1}+2\multsp \Mvariable{u2}\multsp \Mvariable{x2}\multsp
\Mvariable{x5}\multsp \Mvariable{u1}+2\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{x5}\multsp \Mvariable{u1}-  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} {{\Mvariable{x1}}^2}\multsp \Mvariable{x6}\multsp
\Mvariable{u1}+{{\Mvariable{x2}}^2}\multsp \Mvariable{x6}\multsp \Mvariable{u1}-2\multsp \Mvariable{u2}\multsp
\Mvariable{x1}\multsp \Mvariable{x6}\multsp \Mvariable{u1}-2\multsp \Mvariable{u3}\multsp \Mvariable{x2}\multsp
\Mvariable{x6}\multsp
\Mvariable{u1}+\Mvariable{u3}\multsp {{\Mvariable{x1}}^2}\multsp \Mvariable{x5}-  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} \Mvariable{u3}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{x5}-2\multsp
\Mvariable{u2}\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{x5}+\Mvariable{u2}\multsp
{{\Mvariable{x1}}^2}\multsp \Mvariable{x6}-\Mvariable{u2}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{x6}+2\multsp
\Mvariable{u3}\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{x6}\\
\MathEnd{MathArray} }

\dispTFinmath{
\Mvariable{polys}=\Mfunction{Map}[\Mfunction{Numerator}[\Mfunction{Together}[\#]]\&,\Mvariable{Gleichungen}] }

\dispTFoutmath{ \MathBegin{MathArray}[p]{l} \{\Mvariable{u3}\multsp {{\Mvariable{x5}}^3}-3\multsp \Mvariable{u2}\multsp
\Mvariable{x6}\multsp {{\Mvariable{x5}}^2}-3\multsp \Mvariable{u3}\multsp
{{\Mvariable{x6}}^2}\multsp \Mvariable{x5}+\Mvariable{u2}\multsp {{\Mvariable{x6}}^3},  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} -2\multsp \Mvariable{u3}\multsp {{\Mvariable{u1}}^6}+3\multsp
\Mvariable{x2}\multsp {{\Mvariable{u1}}^6}+6\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp
{{\Mvariable{u1}}^5}+6\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp {{\Mvariable{u1}}^5}-12\multsp
\Mvariable{u2}\multsp
\Mvariable{x2}\multsp {{\Mvariable{u1}}^5}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp
{{\Mvariable{u1}}^5}-2\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{u1}}^4}-{{\Mvariable{x2}}^3}\multsp
{{\Mvariable{u1}}^4}-6\multsp \Mvariable{u3}\multsp {{\Mvariable{x1}}^2}\multsp {{\Mvariable{u1}}^4}+6\multsp
\Mvariable{u3}\multsp {{\Mvariable{x2}}^2}\multsp
{{\Mvariable{u1}}^4}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{u3}\multsp
{{\Mvariable{u1}}^4}-18\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{u1}}^4}+18\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^4}+3\multsp
{{\Mvariable{x1}}^2}\multsp
\Mvariable{x2}\multsp {{\Mvariable{u1}}^4}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 24\multsp \Mvariable{u2}\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp
{{\Mvariable{u1}}^4}+2\multsp \Mvariable{u2}\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{u1}}^3}+2\multsp
\Mvariable{u3}\multsp {{\Mvariable{x1}}^3}\multsp {{\Mvariable{u1}}^3}+4\multsp \Mvariable{u2}\multsp
{{\Mvariable{x2}}^3}\multsp
{{\Mvariable{u1}}^3}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 18\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x1}}^2}\multsp {{\Mvariable{u1}}^3}-18\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x2}}^2}\multsp {{\Mvariable{u1}}^3}-6\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{x2}}^2}\multsp
{{\Mvariable{u1}}^3}+2\multsp {{\Mvariable{u2}}^3}\multsp \Mvariable{u3}\multsp {{\Mvariable{u1}}^3}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp {{\Mvariable{u3}}^3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{u1}}^3}+18\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{u1}}^3}-12\multsp {{\Mvariable{u2}}^3}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^3}-12\multsp
\Mvariable{u2}\multsp
{{\Mvariable{x1}}^2}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^3}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 36\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{x1}\multsp
\Mvariable{x2}\multsp {{\Mvariable{u1}}^3}-6\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x1}}^3}\multsp {{\Mvariable{u1}}^2}-6\multsp {{\Mvariable{u2}}^2}\multsp {{\Mvariable{x2}}^3}\multsp
{{\Mvariable{u1}}^2}-6\multsp
{{\Mvariable{u3}}^3}\multsp {{\Mvariable{x1}}^2}\multsp {{\Mvariable{u1}}^2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 18\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x1}}^2}\multsp {{\Mvariable{u1}}^2}+6\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{x2}}^2}\multsp
{{\Mvariable{u1}}^2}+18\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{u3}\multsp {{\Mvariable{x2}}^2}\multsp
{{\Mvariable{u1}}^2}+
 \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 18\multsp \Mvariable{u2}\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{x2}}^2}\multsp {{\Mvariable{u1}}^2}-6\multsp \Mvariable{u2}\multsp {{\Mvariable{u3}}^3}\multsp
\Mvariable{x1}\multsp {{\Mvariable{u1}}^2}-6\multsp {{\Mvariable{u2}}^3}\multsp \Mvariable{u3}\multsp
\Mvariable{x1}\multsp {{\Mvariable{u1}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 3\multsp {{\Mvariable{u2}}^4}\multsp \Mvariable{x2}\multsp
{{\Mvariable{u1}}^2}-3\multsp {{\Mvariable{u3}}^4}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^2}+18\multsp
{{\Mvariable{u2}}^2}\multsp {{\Mvariable{x1}}^2}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^2}+24\multsp
{{\Mvariable{u2}}^3}\multsp
\Mvariable{x1}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{x1}}^3}\multsp
\Mvariable{u1}+6\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{u3}\multsp {{\Mvariable{x1}}^3}\multsp
\Mvariable{u1}+4\multsp {{\Mvariable{u2}}^3}\multsp {{\Mvariable{x2}}^3}\multsp \Mvariable{u1}+6\multsp
\Mvariable{u2}\multsp
{{\Mvariable{u3}}^3}\multsp {{\Mvariable{x1}}^2}\multsp \Mvariable{u1}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp {{\Mvariable{u2}}^3}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x1}}^2}\multsp \Mvariable{u1}-6\multsp \Mvariable{u2}\multsp {{\Mvariable{u3}}^3}\multsp
{{\Mvariable{x2}}^2}\multsp \Mvariable{u1}-6\multsp {{\Mvariable{u2}}^3}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x2}}^2}\multsp
\Mvariable{u1}-6\multsp {{\Mvariable{u3}}^3}\multsp \Mvariable{x1}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{u1}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 18\multsp {{\Mvariable{u2}}^2}\multsp \Mvariable{u3}\multsp
\Mvariable{x1}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{u1}-12\multsp {{\Mvariable{u2}}^3}\multsp
{{\Mvariable{x1}}^2}\multsp \Mvariable{x2}\multsp \Mvariable{u1}-6\multsp {{\Mvariable{u2}}^4}\multsp
\Mvariable{x1}\multsp
\Mvariable{x2}\multsp \Mvariable{u1}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp {{\Mvariable{u3}}^4}\multsp \Mvariable{x1}\multsp
\Mvariable{x2}\multsp \Mvariable{u1}-2\multsp \Mvariable{u2}\multsp {{\Mvariable{u3}}^3}\multsp
{{\Mvariable{x1}}^3}-2\multsp {{\Mvariable{u2}}^3}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x1}}^3}-{{\Mvariable{u2}}^4}\multsp {{\Mvariable{x2}}^3}+{{\Mvariable{u3}}^4}\multsp
{{\Mvariable{x2}}^3}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp \Mvariable{u2}\multsp {{\Mvariable{u3}}^3}\multsp
\Mvariable{x1}\multsp {{\Mvariable{x2}}^2}+6\multsp {{\Mvariable{u2}}^3}\multsp \Mvariable{u3}\multsp
\Mvariable{x1}\multsp {{\Mvariable{x2}}^2}+3\multsp {{\Mvariable{u2}}^4}\multsp {{\Mvariable{x1}}^2}\multsp
\Mvariable{x2}-3\multsp
{{\Mvariable{u3}}^4}\multsp {{\Mvariable{x1}}^2}\multsp \Mvariable{x2},  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} -3\multsp \Mvariable{x4}\multsp {{\Mvariable{u2}}^7}-\Mvariable{u1}\multsp
\Mvariable{u3}\multsp {{\Mvariable{u2}}^6}+3\multsp \Mvariable{u3}\multsp \Mvariable{x3}\multsp
{{\Mvariable{u2}}^6}+3\multsp \Mvariable{u1}\multsp \Mvariable{x4}\multsp {{\Mvariable{u2}}^6}+6\multsp
\Mvariable{x3}\multsp
\Mvariable{x4}\multsp {{\Mvariable{u2}}^6}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} {{\Mvariable{x4}}^3}\multsp {{\Mvariable{u2}}^5}-6\multsp
\Mvariable{u3}\multsp {{\Mvariable{x3}}^2}\multsp {{\Mvariable{u2}}^5}+6\multsp \Mvariable{u3}\multsp
{{\Mvariable{x4}}^2}\multsp {{\Mvariable{u2}}^5}-9\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^5}-3\multsp
{{\Mvariable{x3}}^2}\multsp \Mvariable{x4}\multsp {{\Mvariable{u2}}^5}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp \Mvariable{u1}\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^5}-3\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{u2}}^4}+3\multsp
\Mvariable{u3}\multsp {{\Mvariable{x3}}^3}\multsp {{\Mvariable{u2}}^4}-\Mvariable{u1}\multsp
{{\Mvariable{x4}}^3}\multsp
{{\Mvariable{u2}}^4}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 3\multsp \Mvariable{u1}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x3}}^2}\multsp {{\Mvariable{u2}}^4}-3\multsp \Mvariable{u1}\multsp \Mvariable{u3}\multsp
{{\Mvariable{x4}}^2}\multsp {{\Mvariable{u2}}^4}-9\multsp \Mvariable{u3}\multsp \Mvariable{x3}\multsp
{{\Mvariable{x4}}^2}\multsp
{{\Mvariable{u2}}^4}+9\multsp {{\Mvariable{u3}}^3}\multsp \Mvariable{x3}\multsp {{\Mvariable{u2}}^4}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 9\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^2}\multsp
\Mvariable{x4}\multsp {{\Mvariable{u2}}^4}+3\multsp \Mvariable{u1}\multsp {{\Mvariable{x3}}^2}\multsp
\Mvariable{x4}\multsp {{\Mvariable{u2}}^4}+6\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x3}\multsp
\Mvariable{x4}\multsp
{{\Mvariable{u2}}^4}-2\multsp \Mvariable{u1}\multsp \Mvariable{u3}\multsp {{\Mvariable{x3}}^3}\multsp {{\Mvariable{u2}}^3}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp {{\Mvariable{u3}}^2}\multsp {{\Mvariable{x4}}^3}\multsp
{{\Mvariable{u2}}^3}-12\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{x3}}^2}\multsp {{\Mvariable{u2}}^3}+12\multsp
{{\Mvariable{u3}}^3}\multsp {{\Mvariable{x4}}^2}\multsp {{\Mvariable{u2}}^3}+6\multsp \Mvariable{u1}\multsp
\Mvariable{u3}\multsp
\Mvariable{x3}\multsp {{\Mvariable{x4}}^2}\multsp {{\Mvariable{u2}}^3}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 9\multsp {{\Mvariable{u3}}^4}\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^3}+6\multsp {{\Mvariable{u3}}^2}\multsp {{\Mvariable{x3}}^2}\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^3}-12\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x3}\multsp
\Mvariable{x4}\multsp
{{\Mvariable{u2}}^3}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 3\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^5}\multsp
{{\Mvariable{u2}}^2}+2\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{x3}}^3}\multsp {{\Mvariable{u2}}^2}+6\multsp
\Mvariable{u1}\multsp {{\Mvariable{u3}}^3}\multsp {{\Mvariable{x3}}^2}\multsp {{\Mvariable{u2}}^2}-6\multsp
\Mvariable{u1}\multsp
{{\Mvariable{u3}}^3}\multsp {{\Mvariable{x4}}^2}\multsp {{\Mvariable{u2}}^2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp {{\Mvariable{u3}}^3}\multsp \Mvariable{x3}\multsp
{{\Mvariable{x4}}^2}\multsp {{\Mvariable{u2}}^2}+9\multsp {{\Mvariable{u3}}^5}\multsp \Mvariable{x3}\multsp
{{\Mvariable{u2}}^2}+9\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^4}\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^2}-6\multsp
{{\Mvariable{u3}}^4}\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp {{\Mvariable{u2}}^2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^3}\multsp
{{\Mvariable{x3}}^3}\multsp \Mvariable{u2}-3\multsp {{\Mvariable{u3}}^4}\multsp {{\Mvariable{x4}}^3}\multsp
\Mvariable{u2}-6\multsp {{\Mvariable{u3}}^5}\multsp {{\Mvariable{x3}}^2}\multsp \Mvariable{u2}+6\multsp
{{\Mvariable{u3}}^5}\multsp
{{\Mvariable{x4}}^2}\multsp \Mvariable{u2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 6\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^3}\multsp
\Mvariable{x3}\multsp {{\Mvariable{x4}}^2}\multsp \Mvariable{u2}-3\multsp {{\Mvariable{u3}}^6}\multsp
\Mvariable{x4}\multsp \Mvariable{u2}+9\multsp {{\Mvariable{u3}}^4}\multsp {{\Mvariable{x3}}^2}\multsp
\Mvariable{x4}\multsp
\Mvariable{u2}-6\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^4}\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp \Mvariable{u2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} \Mvariable{u1}\multsp {{\Mvariable{u3}}^7}-{{\Mvariable{u3}}^5}\multsp
{{\Mvariable{x3}}^3}+\Mvariable{u1}\multsp {{\Mvariable{u3}}^4}\multsp {{\Mvariable{x4}}^3}+3\multsp
\Mvariable{u1}\multsp {{\Mvariable{u3}}^5}\multsp {{\Mvariable{x3}}^2}-3\multsp \Mvariable{u1}\multsp
{{\Mvariable{u3}}^5}\multsp
{{\Mvariable{x4}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 3\multsp {{\Mvariable{u3}}^5}\multsp \Mvariable{x3}\multsp
{{\Mvariable{x4}}^2}+3\multsp {{\Mvariable{u3}}^7}\multsp \Mvariable{x3}+3\multsp \Mvariable{u1}\multsp
{{\Mvariable{u3}}^6}\multsp \Mvariable{x4}-3\multsp \Mvariable{u1}\multsp {{\Mvariable{u3}}^4}\multsp
{{\Mvariable{x3}}^2}\multsp
\Mvariable{x4}-6\multsp {{\Mvariable{u3}}^6}\multsp \Mvariable{x3}\multsp \Mvariable{x4},  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} -\Mvariable{x2}\multsp {{\Mvariable{u2}}^3}+2\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^3}+\Mvariable{u3}\multsp \Mvariable{x1}\multsp {{\Mvariable{u2}}^2}-2\multsp \Mvariable{u3}\multsp
\Mvariable{x3}\multsp {{\Mvariable{u2}}^2}+2\multsp \Mvariable{x2}\multsp \Mvariable{x3}\multsp
{{\Mvariable{u2}}^2}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp \Mvariable{x1}\multsp \Mvariable{x4}\multsp
{{\Mvariable{u2}}^2}-2\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp {{\Mvariable{u2}}^2}+2\multsp
\Mvariable{u3}\multsp {{\Mvariable{x3}}^2}\multsp \Mvariable{u2}-\Mvariable{x2}\multsp {{\Mvariable{x3}}^2}\multsp
\Mvariable{u2}-2\multsp \Mvariable{u3}\multsp
{{\Mvariable{x4}}^2}\multsp \Mvariable{u2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} \Mvariable{x2}\multsp {{\Mvariable{x4}}^2}\multsp
\Mvariable{u2}-{{\Mvariable{u3}}^2}\multsp \Mvariable{x2}\multsp \Mvariable{u2}+2\multsp {{\Mvariable{u3}}^2}\multsp
\Mvariable{x4}\multsp \Mvariable{u2}+2\multsp \Mvariable{x1}\multsp \Mvariable{x3}\multsp \Mvariable{x4}\multsp
\Mvariable{u2}-\Mvariable{u3}\multsp
\Mvariable{x1}\multsp {{\Mvariable{x3}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} \Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{x4}}^2}+{{\Mvariable{u3}}^3}\multsp \Mvariable{x1}-2\multsp {{\Mvariable{u3}}^3}\multsp
\Mvariable{x3}+2\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x2}\multsp \Mvariable{x3}-2\multsp
{{\Mvariable{u3}}^2}\multsp \Mvariable{x1}\multsp
\Mvariable{x4}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} 2\multsp {{\Mvariable{u3}}^2}\multsp \Mvariable{x3}\multsp
\Mvariable{x4}-2\multsp \Mvariable{u3}\multsp \Mvariable{x2}\multsp \Mvariable{x3}\multsp
\Mvariable{x4},\Mvariable{x4}\multsp {{\Mvariable{x5}}^2}-2\multsp \Mvariable{x3}\multsp \Mvariable{x6}\multsp
\Mvariable{x5}-\Mvariable{x4}\multsp
{{\Mvariable{x6}}^2},  \\
\noalign{\vspace{0.601563ex}} \hspace{1.em} -\Mvariable{u3}\multsp {{\Mvariable{u1}}^3}+2\multsp \Mvariable{x2}\multsp
{{\Mvariable{u1}}^3}-\Mvariable{x6}\multsp {{\Mvariable{u1}}^3}+2\multsp \Mvariable{u3}\multsp \Mvariable{x1}\multsp
{{\Mvariable{u1}}^2}-2\multsp \Mvariable{u2}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^2}-2\multsp
\Mvariable{x1}\multsp \Mvariable{x2}\multsp {{\Mvariable{u1}}^2}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} \Mvariable{u3}\multsp \Mvariable{x5}\multsp {{\Mvariable{u1}}^2}-2\multsp
\Mvariable{x2}\multsp \Mvariable{x5}\multsp {{\Mvariable{u1}}^2}+\Mvariable{u2}\multsp \Mvariable{x6}\multsp
{{\Mvariable{u1}}^2}+2\multsp \Mvariable{x1}\multsp \Mvariable{x6}\multsp {{\Mvariable{u1}}^2}-\Mvariable{u3}\multsp
{{\Mvariable{x1}}^2}\multsp
\Mvariable{u1}+  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} \Mvariable{u3}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{u1}+2\multsp
\Mvariable{u2}\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{u1}-2\multsp \Mvariable{u3}\multsp
\Mvariable{x1}\multsp \Mvariable{x5}\multsp \Mvariable{u1}+2\multsp \Mvariable{u2}\multsp \Mvariable{x2}\multsp
\Mvariable{x5}\multsp \Mvariable{u1}+2\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{x5}\multsp \Mvariable{u1}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} {{\Mvariable{x1}}^2}\multsp \Mvariable{x6}\multsp
\Mvariable{u1}+{{\Mvariable{x2}}^2}\multsp \Mvariable{x6}\multsp \Mvariable{u1}-2\multsp \Mvariable{u2}\multsp
\Mvariable{x1}\multsp \Mvariable{x6}\multsp \Mvariable{u1}-2\multsp \Mvariable{u3}\multsp \Mvariable{x2}\multsp
\Mvariable{x6}\multsp
\Mvariable{u1}+\Mvariable{u3}\multsp {{\Mvariable{x1}}^2}\multsp \Mvariable{x5}-  \\
\noalign{\vspace{0.601563ex}} \hspace{2.em} \Mvariable{u3}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{x5}-2\multsp
\Mvariable{u2}\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{x5}+\Mvariable{u2}\multsp
{{\Mvariable{x1}}^2}\multsp \Mvariable{x6}-\Mvariable{u2}\multsp {{\Mvariable{x2}}^2}\multsp \Mvariable{x6}+2\multsp
\Mvariable{u3}\multsp \Mvariable{x1}\multsp \Mvariable{x2}\multsp \Mvariable{x6}\}\\
\MathEnd{MathArray} }

\dispTFinmath{ \MathBegin{MathArray}{l}
(*\Mvariable{conclusion}*)  \\
\noalign{\vspace{0.625ex}}\Mvariable{con1}=\Muserfunction{sqrdist}[D,E]-\Muserfunction{sqrdist}[E,F]\\
\MathEnd{MathArray} }

\dispTFoutmath{ -{{\Mvariable{x3}}^2}+2\multsp \Mvariable{x1}\multsp
\Mvariable{x3}-{{\Mvariable{x4}}^2}+{{\Mvariable{x5}}^2}+{{\Mvariable{x6}}^2}+2\multsp \Mvariable{x2}\multsp
\Mvariable{x4}-2\multsp \Mvariable{x1}\multsp \Mvariable{x5}-2\multsp \Mvariable{x2}\multsp \Mvariable{x6} }

\dispTFinmath{ \Mvariable{con2}=\Muserfunction{sqrdist}[D,E]-\Muserfunction{sqrdist}[D,F] }

\dispTFoutmath{ {{\Mvariable{x1}}^2}-2\multsp \Mvariable{x5}\multsp
\Mvariable{x1}+{{\Mvariable{x2}}^2}-{{\Mvariable{x3}}^2}-{{\Mvariable{x4}}^2}+2\multsp \Mvariable{x3}\multsp
\Mvariable{x5}-2\multsp \Mvariable{x2}\multsp \Mvariable{x6}+2\multsp \Mvariable{x4}\multsp \Mvariable{x6} }

\pagebreak
\dispTFinmath{
\Muserfunction{IdealMember}[\Mvariable{con1},\Mvariable{polys},\Mvariable{vars}]//\Mfunction{Timing} }

\Print{
No more memory available.\\
Mathematica kernel has shut down.\\
Try quitting other applications and then retry.}

\dispTFinmath{ \Muserfunction{IdealMember}[\Mvariable{con2},\Mvariable{polys},\Mvariable{vars}]//\Mfunction{Timing} }

\Print{
No more memory available.\\
Mathematica kernel has shut down.\\
Try quitting other applications and then retry.}

Nach einigen Stunden Rechenzeit lief bei \ma\ der Speicher voll und die Berechnung wurde abgebrochen. Magma brauchte
zwar nicht soviel Speicher, aber es war auch nicht in der Lage die Berechnung innerhalb von 12 Stunden zu beenden.

Hier ist der Reduce Code abgedruckt, der zu dem automatischen Beweis f{\"u}hrt (auch zu finden unter
\urlvm{http://vm4.de/diplom/Mathematica/timing/morley-drittel.red}):
\begin{verbatim}
on time;
L1:={
  u3*x5^3 - 3*u2*x5^2*x6 - 3*u3*x5*x6^2 + u2*x6^3,
  -2*u1^6*u3 + 6*u1^5*u2*u3 - 6*u1^4*u2^2*u3 + 2*u1^3*u2^3*
   u3 - 2*u1^4*u3^3 + 2*u1^3*u2*u3^3 + 6*u1^5*u3*x1 -
  18*u1^4*u2*u3*x1 + 18*u1^3*u2^2*u3*x1 -
  6*u1^2*u2^3*u3*x1 + 6*u1^3*u3^3*x1 - 6*u1^2*u2*u3^3*
   x1 - 6*u1^4*u3*x1^2 + 18*u1^3*u2*u3*x1^2 -
  18*u1^2*u2^2*u3*x1^2 + 6*u1*u2^3*u3*x1^2 -
  6*u1^2*u3^3*x1^2 + 6*u1*u2*u3^3*x1^2 +
  2*u1^3*u3*x1^3 - 6*u1^2*u2*u3*x1^3 + 6*u1*u2^2*u3*x1^3 -
  2*u2^3*u3*x1^3 + 2*u1*u3^3*x1^3 - 2*u2*u3^3*x1^3 +
  3*u1^6*x2 - 12*u1^5*u2*x2 + 18*u1^4*u2^2*x2 - 12*u1^3*u2^3*
   x2 + 3*u1^2*u2^4*x2 - 3*u1^2*u3^4*x2 - 6*u1^5*x1*x2 +
  24*u1^4*u2*x1*x2 - 36*u1^3*u2^2*x1*x2 +
  24*u1^2*u2^3*x1*x2 - 6*u1*u2^4*x1*x2 + 6*u1*u3^4*x1*
   x2 + 3*u1^4*x1^2*x2 - 12*u1^3*u2*x1^2*x2 +
  18*u1^2*u2^2*x1^2*x2 - 12*u1*u2^3*x1^2*x2 +
  3*u2^4*x1^2*x2 - 3*u3^4*x1^2*x2 + 6*u1^4*u3*x2^2 -
  18*u1^3*u2*u3*x2^2 + 18*u1^2*u2^2*u3*x2^2 -
  6*u1*u2^3*u3*x2^2 + 6*u1^2*u3^3*x2^2 -
  6*u1*u2*u3^3*x2^2 - 6*u1^3*u3*x1*x2^2 +
  18*u1^2*u2*u3*x1*x2^2 - 18*u1*u2^2*u3*x1*x2^2 +
  6*u2^3*u3*x1*x2^2 - 6*u1*u3^3*x1*x2^2 +
  6*u2*u3^3*x1*x2^2 - u1^4*x2^3 +
  4*u1^3*u2*x2^3 - 6*u1^2*u2^2*
   x2^3 + 4*u1*u2^3*x2^3 - u2^4*x2^3 + u3^4*x2^3,
 -(u1*u2^6*u3) - 3*u1*u2^4*u3^3 - 3*u1*u2^2*u3^5 - u1*u3^7 +
  3*u2^6*u3*x3 + 9*u2^4*u3^3*x3 + 9*u2^2*u3^5*x3 + 3*u3^7*x3 +
  3*u1*u2^4*u3*x3^2 - 6*u2^5*u3*x3^2 + 6*u1*u2^2*u3^3*x3^2 -
  12*u2^3*u3^3*x3^2 + 3*u1*u3^5*x3^2 - 6*u2*u3^5*
   x3^2 - 2*u1*u2^3*u3*x3^3 + 3*u2^4*u3*x3^3 -
   2*u1*u2*u3^3*x3^3 + 2*u2^2*u3^3*x3^3 - u3^5*x3^3 +
   3*u1*u2^6*x4 - 3*u2^7*x4 + 9*u1*u2^4*u3^2*x4 -
  9*u2^5*u3^2*x4 + 9*u1*u2^2*u3^4*x4 - 9*u2^3*u3^4*x4 +
  3*u1*u3^6*x4 - 3*u2*u3^6*x4 - 6*u1*u2^5*x3*x4 + 6*u2^6*x3*
   x4 - 12*u1*u2^3*u3^2*x3*x4 + 6*u2^4*u3^2*x3*x4 -
  6*u1*u2*u3^4*x3*x4 - 6*u2^2*u3^4*x3*x4 -
  6*u3^6*x3*x4 + 3*u1*u2^4*x3^2*x4 - 3*u2^5*x3^2*x4 +
  6*u2^3*u3^2*x3^2*x4 - 3*u1*u3^4*x3^2*x4 +
  9*u2*u3^4*x3^2*x4 - 3*u1*u2^4*u3*x4^2 +
  6*u2^5*u3*x4^2 - 6*u1*u2^2*u3^3*x4^2 + 12*u2^3*u3^3*x4^2 -
  3*u1*u3^5*x4^2 + 6*u2*u3^5*x4^2 + 6*u1*u2^3*u3*x3*x4^2 -
  9*u2^4*u3*x3*x4^2 + 6*u1*u2*u3^3*x3*x4^2 -
  6*u2^2*u3^3*x3*x4^2 + 3*u3^5*x3*x4^2 - u1*u2^4*x4^3 +
  u2^5*x4^3 - 2*u2^3*u3^2*x4^3 + u1*u3^4*x4^3 - 3*u2*u3^4*x4^3,
 u2^2*u3*x1 + u3^3*x1 - u2^3*x2 - u2*u3^2*x2 - 2*u2^2*u3*x3 -
  2*u3^3*x3 + 2*u2^2*x2*x3 + 2*u3^2*x2*x3 + 2*u2*u3*x3^2 -
  u3*x1*x3^2 - u2*x2*x3^2 + 2*u2^3*x4 + 2*u2*u3^2*x4 -
  2*u2^2*x1*x4 - 2*u3^2*x1*x4 - 2*u2^2*x3*x4 + 2*u3^2*x3*x4 +
  2*u2*x1*x3*x4 - 2*u3*x2*x3*x4 - 2*u2*u3*x4^2 + u3*x1*x4^2 +
  u2*x2*x4^2, x4*x5^2 - 2*x3*x5*x6 - x4*x6^2,
 -(u1^3*u3) + 2*u1^2*u3*x1 - u1*u3*x1^2 + 2*u1^3*x2 -
  2*u1^2*u2*x2 - 2*u1^2*x1*x2 + 2*u1*u2*x1*x2 + u1*u3*x2^2 +
  u1^2*u3*x5 - 2*u1*u3*x1*x5 + u3*x1^2*x5 - 2*u1^2*x2*x5 +
  2*u1*u2*x2*x5 + 2*u1*x1*x2*x5 - 2*u2*x1*x2*x5 - u3*x2^2*x5 -
  u1^3*x6 + u1^2*u2*x6 + 2*u1^2*x1*x6 - 2*u1*u2*x1*x6 -
  u1*x1^2*x6 + u2*x1^2*x6 - 2*u1*u3*x2*x6 + 2*u3*x1*x2*x6 +
  u1*x2^2*x6 - u2*x2^2*x6
};
b:=2*x1*x3 - x3^2 + 2*x2*x4 - x4^2 -2*x1*x5 + x5^2 -
   2*x2*x6 + x6^2;
b2:=x1^2 + x2^2 - x3^2 - x4^2 - 2*x1*x5 + 2*x3*x5 -
    2*x2*x6 + 2*x4*x6;
load groebner;
L2:={1-y*b} . L1;
L3:={1-y*b2} . L1;
{groebner(L2,{x1,x2,x3,x4,x5,x6}),groebner(L3,{x1,x2,x3,x4,x5,x6})};
\end{verbatim}

\end{chapter}
