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/(4*a)=0 holds z=3-root(d/(16*a)+sqrt((d/(16*a))^2+(c/(12*a)) |^ 3)) +
  3-root(d/(16*a)-sqrt((d/(16*a))^2+(c/(12*a)) |^ 3))+ sqrt(3*(3-root(d/(16*a)+
sqrt((d/(16*a))^2+(c/(12*a)) |^ 3)) + 3-root(d/(16*a)-sqrt((d/(16*a))^2+(c/(12*
a)) |^ 3)))^2+c/a)*<i> or z=3-root(d/(16*a)+sqrt((d/(16*a))^2+(c/(12*a)) |^ 3))
+3-root(d/(16*a)-sqrt((d/(16*a))^2+(c/(12*a)) |^ 3)) -sqrt(3*(3-root(d/(16*a)+
sqrt((d/(16*a))^2+(c/(12*a)) |^ 3)) +3-root(d/(16*a)-sqrt((d/(16*a))^2+(c/(12*a
)) |^ 3)))^2+c/a)*<i> or z=2*(3-root(d/(16*a)+sqrt((d/(16*a))^2+(c/(12*a)) |^ 3
)))+ sqrt(3*(2*(3-root(d/(16*a)+sqrt((d/(16*a))^2 +(c/(12*a)) |^ 3))))^2+c/a)*
<i> or z=2*(3-root(d/(16*a)+sqrt((d/(16*a))^2+(c/(12*a)) |^ 3))) -sqrt(3*(2*(3
-root(d/(16*a)+sqrt((d/(16*a))^2 +(c/(12*a)) |^ 3))))^2+c/a)*<i> or z=2*(3-root
(d/(16*a)-sqrt((d/(16*a))^2+(c/(12*a)) |^ 3)))+ sqrt(3*(2*(3-root(d/(16*a)-sqrt
((d/(16*a))^2 +(c/(12*a)) |^ 3))))^2+c/a)*<i> or z=2*(3-root(d/(16*a)-sqrt((d/(
16*a))^2+(c/(12*a)) |^ 3))) -sqrt(3*(2*(3-root(d/(16*a)-sqrt((d/(16*a))^2 +(c/(
  12*a)) |^ 3))))^2+c/a)*<i>
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>;
A5: 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>) +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 - 3*x*y^2)+c*x+d+(a*(-y|^ 3+3*x^2*y) +c*y+0)*<i>;
  then a*(-y|^(2+1)+3*x^2*y)+c*y = 0 by COMPLEX1:4,12;
  then a*(-y|^2*y+3*x^2*y)+c*y = 0 by NEWTON:6;
  then (a*(-y|^2+3*x^2)+c+0)*y = 0;
  then (a*(y|^2))+(-a*(y|^2)+(a*(3*x^2)+c))=(a*(y|^2))+0 by A2,XCMPLX_1:6;
  then y|^(1+1) = (a*(3*x^2)+c)/a by A1,XCMPLX_1:89;
  then y|^1*y = (a*(3*x^2)+c)/a by NEWTON:6;
  then
A6: y^2 = (3*x^2)*a/a+c/a+0/a;
  then
A7: z = x+y*<i> & y^2 = 3*x^2+c/a+0*a" by A1,COMPLEX1:13,XCMPLX_1:89;
  set q = -d/(8*a);
  set pp = c/(4*a);
  let u,v;
  set m = 3-root(-q/2+sqrt(q^2/4+(pp/3) |^ 3));
  set n = 3-root(-q/2-sqrt(q^2/4+(pp/3) |^ 3));
A8: (c/(4*a))/3=1/3*(c/(4*a)) .=c/(a*4*3) by XCMPLX_1:103
    .=c/(a*(4*3));
  -q/2=1/2*(d/(8*a));
  then
A9: -q/2=d/(a*8*2) by XCMPLX_1:103;
  a*(-y|^ 3+3*x^2*y) +c*y+0 = 0 by A5,COMPLEX1:4,12;
  then a*(x|^ 3 - 3*x*(3*x^2+c/a)+0)+c*x+d = 0 by A1,A5,A6,XCMPLX_1:89;
  then 0 =a*x|^ 3 - a*(9*(x*x^2)+(3*x)*(c/a))+c*x+d
    .=a*x|^ 3 - a*(9*(x|^1*x*x)+(3*x)*(c/a))+c*x+d
    .=a*x|^ 3 - a*(9*(x|^(1+1)*x)+(3*x)*(c/a))+c*x+d by NEWTON:6
    .=a*x|^ 3 - a*(9*(x|^(2+1))+(3*x)*(c/a)+0)+c*x+d by NEWTON:6
    .=a*x|^ 3 - (a*(9*(x|^3))+(3*x)*(c*(a/a)))+c*x+d
    .=a*x|^ 3 - (a*(9*(x|^3))+(3*x)*c)+c*x+d by A1,XCMPLX_1:88
    .=(-8*a)*x|^3+(-2*c)*x+d;
  then (-1)*0=(8*a)*x|^3+(2*c)*x+(-d);
  then 0=x|^3*((8*a)/(8*a))+(8*a)"*((2*c)*x)+(8*a)"*(-d);
  then 0=1*x|^3+0*x^2+((2*c)/(8*a))*x+((8*a)"*(-d)) by A1,XCMPLX_1:88;
  then 0=1*x|^3+0*x^2+((2/8*c)/a)*x+((-d)/(8*a)) by XCMPLX_1:83;
  then 0=1*x|^3+0*x^2+(1/a*(c/4))*x+((-d)/(8*a));
  then
A10: Polynom(1,0,c/(4*a),-d/(8*a),x) = 0 by XCMPLX_1:103;
  assume Re z=u+v & 3*v*u+c/(4*a)=0;
  then
A11: x = 3-root(-q/2+sqrt(q^2/4+(pp/3) |^ 3)) + 3-root(-q/2-sqrt(q^2/4+(pp/3
) |^ 3)) or x = 3-root(-q/2+sqrt(q^2/4+(pp/3) |^ 3)) + 3-root(-q/2+sqrt(q^2/4+(
pp/3) |^ 3)) or x = 3-root(-q/2-sqrt(q^2/4+(pp/3) |^ 3)) + 3-root(-q/2-sqrt(q^2
  /4+(pp/3) |^ 3)) by A10,POLYEQ_1:19;
A12: now
    per cases by A11;
    case
      x = m + n;
      hence
      z=m+n+sqrt(3*(m + n )^2+c/a)*<i> or z=m+n+(-sqrt(3*(m + n )^2+c/a))
      *<i> by A7,Th12;
    end;
    case
      x = 2*m;
      hence z=2*m+sqrt(3*(2*m)^2+c/a)*<i> or z=2*m+(-sqrt(3*(2*m)^2+c/a))*<i>
      by A7,Th12;
    end;
    case
      x = 2*n;
      hence
      z=2*n+sqrt(3*(2*n )^2+c/a)*<i> or z=2*n+(-sqrt(3*(2*n )^2+c/a))*<i>
      by A7,Th12;
    end;
  end;
  q^2/4=(1/2*(d/(8*a)))^2;
  hence thesis by A9,A12,A8;
end;
