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 & p = c/a & q = d/a & Polynom(a,0,c,d,x) = 0 implies for u,v st
x = u+v & 3*v*u+p = 0 holds x = 3-root(-d/(2*a)+sqrt((d^2/(4*a^2))+(c/(3*a)) |^
3)) +3-root(-d/(2*a)-sqrt((d^2/(4*a^2))+(c/(3*a)) |^ 3)) or x = 3-root(-d/(2*a)
+sqrt((d^2/(4*a^2))+(c/(3*a)) |^ 3)) +3-root(-d/(2*a)+sqrt((d^2/(4*a^2))+(c/(3*
a)) |^ 3)) or x = 3-root(-d/(2*a)-sqrt((d^2/(4*a^2))+(c/(3*a)) |^ 3)) +3-root(-
  d/(2*a)-sqrt((d^2/(4*a^2))+(c/(3*a)) |^ 3))
proof
  assume that
A1: a <> 0 and
A2: p = c/a and
A3: q = d/a;
A4: p/3 = c/(3*a) & -q/2 = -d/(2*a) by A2,A3,XCMPLX_1:78;
  assume Polynom(a,0,c,d,x) = 0;
  then a"*(a*(x |^ 3)+c*x +d) = 0;
  then (a"*a)*(x |^ 3)+a"*(c*x +d)= 0;
  then 1*(x |^ 3)+a"*(c*x +d)= 0 by A1,XCMPLX_0:def 7;
  then (x |^ 3)+((a"*c)*x +a"*d) = 0;
  then (x |^ 3)+((c/a)*x +a"*d) = 0 by XCMPLX_0:def 9;
  then (x |^ 3)+((c/a)*x +d/a) = 0 by XCMPLX_0:def 9;
  then
A5: Polynom(1,0,p,q,x) = 0 by A2,A3;
  q^2/4 = d^2/a^2/4 by A3,XCMPLX_1:76;
  then
A6: q^2/4 = d^2/(4*a^2) by XCMPLX_1:78;
  let u,v;
  assume x = u+v & 3*v*u+p = 0;
  hence thesis by A5,A4,A6,Th19;
end;
