
theorem Th10:
  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^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
proof
  let a1,a2,a3,a4,a5,x,x1,x2,x3,x4 be Real;
  assume
A1: a1 <> 0;
  set z = ((x-x1)*(x-x2)*(x-x3)*(x-x4));
  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 Polynom(a1,a2,a3,a4,a5,x) = Four0(a1,x1,x2,x3,x4,x);
  then w/a1*a1-(z*a1) = (z*a1)-(z*a1) by A1,XCMPLX_1:87;
  then (w/a1-z)*a1 = 0;
  then w/a1+-z= 0-0 by A1,XCMPLX_1:6;
  hence thesis;
end;
