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
  a <> 0 & delta(a,b,c) >= 0 & Polynom(a,b,c,0,x) = 0 implies x = 0 or x
  = (-b+sqrt delta(a,b,c))/(2*a) or x = (-b-sqrt delta(a,b,c))/(2*a)
proof
  assume
A1: a <> 0 & delta(a,b,c)>= 0;
  x |^ 3 = x |^ (2+1);
  then x |^ 3 = (x |^ (1+1))*x by NEWTON:6;
  then
A2: x |^ 3 = ((x |^ 1)*x)*x by NEWTON:6;
A3: x |^ 3 = x^2*x by A2;
  assume Polynom(a,b,c,0,x) = 0;
  then (a*x^2 +b*x+c)*x = 0 by A3;
  then x = 0 or Polynom(a,b,c,x) = 0 by XCMPLX_1:6;
  hence thesis by A1,Th5;
end;
