reserve a,b,c,d,a9,b9,c9,d9,y,x1,u,v for Real,
  s,t,h,z,z1,z2,z3,s1,s2,s3 for Complex;

theorem
  for a,b,c being Real,z,z1,z2 being Complex st a <> 0 & for z
being Complex holds Polynom(a,b,c,z) = Quard(a,z1,z2,z) holds b/a =-(z1+
  z2) & c/a = z1*z2
proof
  let a,b,c be Real,z,z1,z2 be Complex;
  assume
A1: a <> 0;
  assume
A2: for z being Complex holds Polynom(a,b,c,z) = Quard(a,z1,z2,z);
  then
A3: Polynom(a,b,c,0) = Quard(a,z1,z2,0);
  Quard(a,z1,z2,1) = Polynom(a,b,c,1) by A2
    .=a+b+c;
  then a+b+c =a+a*(-(z1+z2))+c by A3;
  hence thesis by A1,A3,XCMPLX_1:203;
end;
