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 & Im z = 0 & Polynom(a,0,c,d,z)=0 implies for u,v st Re z = u+v &
3*v*u+c/a=0 holds z = 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 z = 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 z =
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
A1: a <> 0;
  set y=Im z;
  set x=Re z;
  assume that
A2: Im z = 0 and
A3: Polynom(a,0,c,d,z)=0;
A4: a = a+0*<i>;
  0 = a*(Re z^3+(Im z^3)*<i>)+0*z^2+c*z+d by A3,COMPLEX1:13
    .= a*(((Re z)|^ 3 - 3*Re z*(Im z)^2)+(Im z^3)*<i>) +0*z^2+c*z+d by Th5
    .= a*(((Re z)|^ 3 - 3*Re z*(Im z)^2)+(-(Im z)|^ 3+3*(Re z)^2 *Im z)*<i>)
  +c*z+d by Th5
    .= (a+0*<i>)*((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>) +c*(Re z+Im z*<i>)
  +d by COMPLEX1:13
    .=Re a *Re((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>)-Im a *Im((x|^ 3 - 3*x
*y^2)+(-y|^ 3+3*x^2*y)*<i>)+(Re a *Im((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>)+
  Re((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>)*Im a)*<i> +c*(Re z+Im z*<i>)+d by
COMPLEX1:82
    .=(Re a *(x|^ 3 - 3*x*y^2)-Im a *Im((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*
<i>))+(Re a *Im((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>)+ Re((x|^ 3 - 3*x*y^2)+(
  -y|^ 3+3*x^2*y)*<i>)*Im a )*<i> +c*(Re z+Im z *<i>)+d by COMPLEX1:12
    .=(Re a *(x|^ 3 - 3*x*y^2)-Im a*(-y|^ 3+3*x^2*y))+(Re a *Im((x|^ 3 - 3*x
*y^2)+(-y|^ 3+3*x^2*y)*<i>)+ Re((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>)*Im a )*
  <i> +c*(Re z+Im z *<i>)+d by COMPLEX1:12
    .=(Re a *(x|^ 3 - 3*x*y^2)-Im a*(-y|^ 3+3*x^2*y))+(Re a *Im((x|^ 3 - 3*x
*y^2)+(-y|^ 3+3*x^2*y)*<i>)+ (x|^ 3 - 3*x*y^2)*Im a )*<i> +c*(Re z+Im z *<i>)+d
  by COMPLEX1:12
    .=(Re a *(x|^ 3 - 3*x*y^2)-Im a *(-y|^ 3+3*x^2*y))+(Re a*(-y|^ 3+3*x^2*y
  )+ (x|^ 3 - 3*x*y^2)*Im a )*<i> +c*(Re z+Im z *<i>)+d by COMPLEX1:12
    .=(a*(x|^ 3 - 3*x*y^2)-Im a *(-y|^ 3+3*x^2*y))+(Re a*(-y|^ 3+3*x^2*y)+ (
  x|^ 3 - 3*x*y^2)*Im a )*<i> +c*(Re z+Im z *<i>)+d by A4,COMPLEX1:12
    .=(a*(x|^ 3 - 3*x*y^2)-0*(-y|^ 3+3*x^2*y))+( Re a*(-y|^ 3+3*x^2*y)+(x|^
  3 - 3*x*y^2)*Im a )*<i> +c*(Re z+Im z *<i>)+d by A4,COMPLEX1:12
    .=(a*(x|^ 3 - 3*x*y^2)-0)+(a*(-y|^ 3+3*x^2*y) +(x|^ 3 - 3*x*y^2)*Im a )*
  <i> +c*(Re z+Im z *<i>)+d by A4,COMPLEX1:12
    .=(a*(x|^ 3 - 3*x*y^2)-0)+(a*(-y|^ 3+3*x^2*y) +(x|^ 3 - 3*x*y^2)*0 )*<i>
  +c*(Re z+Im z *<i>)+d by A4,COMPLEX1:12
    .= a*(x|^ 3 - 0)+c*x+d+(a*(-0))*<i> by A2,NEWTON:11
    .= a*x|^ 3 +c*x+d;
  then a"*(a*x|^ 3+c*x+d) = a"*0;
  then x|^ 3*(a/a)+a"*c*x+a"*d = 0;
  then
A5: Polynom(1,0,c/a,d/a,x) = 0 by A1,XCMPLX_1:88;
A6: (d/a)^2/4 =1/4*(d^2/a^2) by XCMPLX_1:76
    .=d^2/(a^2*4) by XCMPLX_1:103;
  let u,v;
  assume
A7: Re z=u+v & 3*v*u+c/a=0;
A8: -(d/a)/2 = -1/2*(d/a) .= -d/(a*2) by XCMPLX_1:103;
A9: (c/a)/3 =1/3*(c/a) .=c/(a*3) by XCMPLX_1:103;
  z = Re z+(Im z)*<i> by COMPLEX1:13
    .= x by A2;
  hence thesis by A7,A5,A8,A6,A9,POLYEQ_1:19;
end;
