reserve a, a9, a1, a2, a3, b, b9, c, c9, d, d9, h, p, q, x, x1, x2, x3, u, v,
  y, z for Real;

theorem
  for a, b, c being Complex holds a <> 0 & (for x being Real
 holds Polynom(a,b,c,x) = Quard(a,x1,x2,x)) implies b/a = -(x1+x2) & c/a
  = x1*x2
proof
  let a, b, c be Complex;
  assume
A1: a <> 0;
  assume
A2: for x being Real holds Polynom(a,b,c,x) = Quard(a,x1,x2,x);
  now
    let z be Real;
    set h = z-x1, t = z-x2;
    set e = h*t-z^2;
    Polynom(a,b,c,z) = Quard(a,x1,x2,z) by A2;
    then a*(h*t-z^2) = (b*z+c);
    then e = (b*z+c)/a by A1,XCMPLX_1:89
      .= a"*(b*z+c) by XCMPLX_0:def 9
      .= a"*(b*z)+a"*c;
    then z*z-z*x2-x1*z+x1*x2 = z^2+(a"*b)*z+a"*c;
    then z^2-(x1+x2)*z+x1*x2 = z^2+(b/a)*z+a"*c by XCMPLX_0:def 9
      .= z^2+(b/a)*z+(c/a) by XCMPLX_0:def 9;
    hence Polynom(1,-(x1+x2),x1*x2,z) = Polynom(1,b/a,c/a,z);
  end;
  hence thesis by Th4;
end;
