
theorem
  for a1,a2,a3,a4,a5,x1,x2,x3,x4 being Real st a1 <> 0 & (for x
  being Real holds Polynom(a1,a2,a3,a4,a5,x) = Four0(a1,x1,x2,x3,x4,x))
holds a2/a1 = -(x1+x2+x3+x4) & a3/a1 = (x1*x2+x1*x3+x1*x4)+(x2*x3+x2*x4)+x3*x4
  & a4/a1 = -(x1*x2*x3+x1*x2*x4+x1*x3*x4+x2*x3*x4) & a5/a1 = x1*x2*x3*x4
proof
  set b1 = 1;
  let a1,a2,a3,a4,a5,x1,x2,x3,x4 be Real;
  assume
A1: a1 <> 0;
  set b5 = x1*x2*x3*x4;
  set b4 = -(x1*x2*x3+x1*x2*x4+x1*x3*x4+x2*x3*x4);
  set b3 = (x1*x2+x1*x3+x1*x4)+(x2*x3+x2*x4)+x3*x4;
  set b2 = -(x1+x2+x3+x4);
  assume
A2: for x being Real holds Polynom(a1,a2,a3,a4,a5,x) = Four0(a1,
  x1,x2,x3,x4,x);
  now
    let x be Real;
    set t= b1*(x|^ 4)+b2*x|^ 3+b3*x^2+b4*x+b5;
    (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) by A1,A2,Th11;
    then t = a1"*((a1*(x|^ 4)+a2*(x|^ 3))+(a3*x^2+a4*x)+a5) by XCMPLX_0:def 9
      .= (a1"*a1)*(x|^ 4)+a1"*(a2*(x|^ 3)) +(a1"*(a3*x^2)+a1"*(a4*x)+a1"*a5)
      .= (a1/a1)*(x|^ 4)+a1"*(a2*(x|^ 3)) +(a1"*(a3*x^2)+a1"*(a4*x)+a1"*a5)
    by XCMPLX_0:def 9
      .= 1 *(x|^ 4)+a1"*(a2*(x|^ 3)) +(a1"*(a3*x^2)+a1"*(a4*x)+a1"*a5) by A1,
XCMPLX_1:60
      .= (x|^ 4)+(a1"*a2)*(x|^ 3) +(a1"*(a3*x^2)+a1"*(a4*x)+a1"*a5)
      .= (x|^ 4)+(a2/a1)*(x|^ 3) +((a1"*a3)*x^2+a1"*(a4*x)+a1"*a5) by
XCMPLX_0:def 9
      .= (x|^ 4)+(a2/a1)*(x|^ 3) +((a3/a1)*x^2+(a1"*a4)*x+a1"*a5) by
XCMPLX_0:def 9
      .= (x|^ 4)+(a2/a1)*(x|^ 3) +((a3/a1)*x^2+(a4/a1)*x+(a1"*a5)) by
XCMPLX_0:def 9
      .= (x|^ 4)+(a2/a1)*(x|^ 3) +((a3/a1)*x^2+(a4/a1)*x+(a5/a1)) by
XCMPLX_0:def 9
      .= Polynom(1,a2/a1,a3/a1,a4/a1,a5/a1,x);
    hence Polynom(1,a2/a1,a3/a1,a4/a1,a5/a1,x) = Polynom(b1,b2,b3,b4,b5,x);
  end;
  hence thesis by Th9;
end;
