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, x being Complex holds a <> 0 & delta(a,b,c) = 0 &
  Polynom(a,b,c,x) = 0 implies x = -(b/(2*a))
proof
  let a, b, c, x be Complex;
  assume that
A1: a <> 0 and
A2: delta(a,b,c) = 0;
  set y = x;
  set t = a^2*y^2+(a*b)*y;
A3: b^2-4*a*c = delta(a,b,c) by QUIN_1:def 1;
  assume Polynom(a,b,c,y) = 0;
  then a*(a*y^2+b*y+c) = 0;
  then t +a*c = 0;
  then (a*y)^2+ 2*((a*y)*b*2") + (b/2)^2 = 0 by A2,A3;
  then (a*y+b/2)^2 = 0;
  then a*y+b/2 = 0 by XCMPLX_1:6;
  then y = (- b*2")/a by A1,XCMPLX_1:89
    .= (-1)*(b*2")*a" by XCMPLX_0:def 9
    .= -(b*(2"*a"))*1
    .= -(b*(2*a)") by XCMPLX_1:204;
  hence thesis by XCMPLX_0:def 9;
end;
