
theorem Th11:
  for a1,a2,a3,a4,a5,x,x1,x2,x3,x4 being Real st a1 <> 0
holds (for x being Real holds Polynom(a1,a2,a3,a4,a5,x) = Four0(a1,x1,x2
,x3,x4,x)) implies (a1*(x|^ 4)+a2*(x|^ 3)+a3*x^2+a4*x+a5)/a1 = x|^ 4-(x1+x2+x3+
x4)*x|^ 3 +((x1*x2+x1*x3+x1*x4)+(x2*x3+x2*x4)+x3*x4)*x^2 -(x1*x2*x3+x1*x2*x4+x1
  *x3*x4+x2*x3*x4)*x+(x1*x2*x3*x4)
proof
  let a1,a2,a3,a4,a5,x,x1,x2,x3,x4 being Real;
  assume
A1: a1 <> 0;
  set w = a1*(x|^ 4)+a2*(x|^ 3)+a3*x^2+a4*x+a5;
  assume for x being Real holds Polynom(a1,a2,a3,a4,a5,x) = Four0(a1,
  x1,x2,x3,x4,x);
  then
  (a1*(x|^ 4)+a2*(x|^ 3)+a3*x^2+a4*x+a5)/a1 = x^2*x^2-(x1+x2+x3)*(x^2*x)+(
  x1*x3+x2*x3+x1*x2)*x^2 -(x1*x2*x3)*x-((x-x1)*(x-x2)*(x-x3))*x4 by A1,Th10;
  then
  w/a1 = x^2*x^2-(x1+x2+x3+x4)*(x^2*x) +(x1*x3+x2*x3+x1*x2+(x2*x4+x1*x4+x3
  *x4))*x^2 -(x1*x2*x3+x1*x2*x4+-(-x1*x3*x4)+x2*x3*x4)*x+(x1*x2*x3*x4);
  then
  w/a1 = x|^ 4-(x1+x2+x3+x4)*(x^2*x) +((x1*x2+x1*x3+x1*x4)+(x2*x3+x2*x4)+
  x3*x4)*x^2 -(x1*x2*x3+x1*x2*x4+x1*x3*x4+x2*x3*x4)*x+(x1*x2*x3*x4) by Th4;
  hence thesis by Th4;
end;
