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
  a<>0 & 4*a*c <= 0 & Polynom(a,0,c,0,z)=0 implies z = (sqrt -4*a*c)/(2*
  a) or z = (-sqrt -4*a*c)/(2*a) or z = 0
proof
  assume that
A1: a<>0 & 4*a*c <= 0 and
A2: Polynom(a,0,c,0,z)=0;
  (a*z^2+c)*z = 0 by A2;
  then Polynom(a,0,c,z) = 0 or z = 0;
  then
  z = (-0+sqrt delta(a,0,c))/(2*a) or z = (-0-sqrt delta(a,0,c))/(2*a) or
  z = 0 or z = 0/(2*a) by A1,Th1;
  hence thesis;
end;
