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
  delta(a,b,c) < 0 implies not ex x st Polynom(a,b,c,x) = 0
proof
  set e = a*c;
  assume delta(a,b,c) < 0;
  then (b^2-4*a*c) < 0 by QUIN_1:def 1;
  then
A1: (b^2-4*(a*c))*4" < 0 by XREAL_1:132;
  given y such that
A2: Polynom(a,b,c,y) = 0;
  set t = a^2*y^2+(a*b)*y;
  a*(a*y^2+b*y+c) = a*0 by A2;
  then t +b^2/4-(b^2*4"-(4*e)*4") = 0;
  then
A3: (a*y+b/2)^2 = (b^2-4*(a*c))*4";
  then (a*y+b/2) > 0 by A1,XREAL_1:133;
  hence contradiction by A3,A1,XREAL_1:133;
end;
