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 & Polynom(a,b,c,d,z)=0 & Im z=0 implies for u,v,x1 st x1=Re z + b
/(3*a) & Re z=u+v-b/(3*a) & 3*u*v+(3*a*c-b^2)/(3*a^2)=0 holds z =3-root(-(b/(3*
a)) |^ 3-(3*a*d-b*c)/(6*a^2) +sqrt((2*((b/(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2))^2/4
  +((3*a*c-b^2)/(9*a^2)) |^ 3)) + 3-root(-(b/(3*a)) |^ 3-(3*a*d-b*c)/(6*a^2) -
sqrt((2*((b/(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2))^2/4 +((3*a*c-b^2)/(9*a^2)) |^ 3))
-b/(3*a)+0*<i> or z =3-root(-(b/(3*a)) |^ 3-(3*a*d-b*c)/(6*a^2) +sqrt((2*((b/(3
*a)) |^ 3)+(3*a*d-b*c)/(3*a^2))^2/4 +((3*a*c-b^2)/(9*a^2)) |^ 3)) + 3-root(-(b/
(3*a)) |^ 3-(3*a*d-b*c)/(6*a^2) +sqrt((2*((b/(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2))
^2/4 +((3*a*c-b^2)/(9*a^2)) |^ 3))-b/(3*a)+0*<i> or z =3-root(-(b/(3*a)) |^ 3-(
3*a*d-b*c)/(6*a^2) -sqrt((2*((b/(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2))^2/4 +((3*a*c-
b^2)/(9*a^2)) |^ 3)) + 3-root(-(b/(3*a)) |^ 3-(3*a*d-b*c)/(6*a^2) -sqrt((2*((b/
(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2))^2/4 +((3*a*c-b^2)/(9*a^2)) |^ 3))-b/(3*a)+0*
  <i>
proof
  assume
A1: a<>0;
  set p = (3*a*c-b^2)/(3*a^2);
  set b9=c/a;
A2: a = a+0*<i>;
  set q = 2*((b/(3*a)) |^ 3)+(3*a*d-b*c)/(3*a^2);
  set c9=d/a;
  set a9=b/a;
  set y=Im z;
  set x=Re z;
  assume that
A3: Polynom(a,b,c,d,z)=0 and
A4: Im z=0;
A5: z = x + y*<i> by COMPLEX1:13;
  0 = a*(Re z^3+(Im z^3)*<i>)+b*z^2+c*z+d by A3,COMPLEX1:13
    .= a*(((Re z)|^ 3 - 3*Re z*(Im z)^2)+(Im z^3)*<i>) +b*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>)
  +b*z^2+c*z+d by Th5
    .= (a+0*<i>)*((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>) +b*z^2+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> +b*z^2+c*(Re z+Im z *<i>)
  +d by COMPLEX1:82
    .= (Re a*Re((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>) -0*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> +b*z^2+c*(Re z+Im z *<i>)
  +d by A2,COMPLEX1:12
    .= (Re a*Re((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>) -0)+(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>)*0
  )*<i> +b*z^2+c*(Re z+Im z *<i>)+d by A2,COMPLEX1:12
    .= (Re a*Re((x|^ 3 - 3*x*y^2)+(-y|^ 3+3*x^2*y)*<i>) -0)+(Re a*(-y|^ 3+3*
  x^2*y)+0 )*<i> +b*z^2+c*(Re z+Im z *<i>)+d by COMPLEX1:12
    .= (Re a*(x|^ 3 - 3*x*y^2)-0)+(Re a*(-y|^ 3+3*x^2*y))*<i> +b*z^2+c*(Re z
  +Im z *<i>)+d by COMPLEX1:12
    .= (a*(x|^ 3 - 3*x*y^2)-0)+(Re a*(-y|^ 3+3*x^2*y))*<i> +b*z^2+c*(Re z+Im
  z *<i>)+d by A2,COMPLEX1:12
    .= a*(x|^ 3 - 3*x*y^2)+(a*(-y|^ 3+3*x^2*y))*<i> +b*(x^2-y^2+(2*x*y)*<i>)
  +(c*x+(c*y)*<i>)+d by A2,A5,COMPLEX1:12;
  then 0 = a*(x|^ 3 - 3*x*0)+b*(x^2-0)+c*x+d +(a*(-0|^ 3+0)+b*0+0)*<i> by A4
    .= (a*x|^ 3+b*x^2+c*x+d)+(a*0+0)*<i> by NEWTON:11;
  then
A6: Polynom(a,b,c,d,x) = 0;
A7: c9=d/a;
  p/3=1/3*((3*a*c-b^2)/(3*a^2));
  then
A8: p/3=(3*a*c-b^2)/(a^2*3*3) by XCMPLX_1:103;
A9: -q/2=-((b/(3*a)) |^ 3+1/2*((3*a*d-b*c)/(3*a^2)))
    .=-((b/(3*a)) |^ 3+(3*a*d-b*c)/(a^2*3*2)) by XCMPLX_1:103
    .=-(b/(3*a)) |^ 3-(3*a*d-b*c)/(6*a^2);
  let u,v,x1;
  assume that
A10: x1=Re z +b/(3*a) & Re z=u+v-b/(3*a) and
A11: 3*u*v+(3*a*c-b^2)/(3*a^2)=0;
  a9=b/a & b9=c/a;
  then Polynom(1,0,p,q,x1) = 0 by A1,A10,A6,A7,POLYEQ_1:16;
  then
  x1 = 3-root(-q/2+sqrt(q^2/4+(p/3) |^ 3)) + 3-root(-q/2-sqrt(q^2/4+(p/3)
|^ 3)) or x1 = 3-root(-q/2+sqrt(q^2/4+(p/3) |^ 3)) + 3-root(-q/2+sqrt(q^2/4+(p/
3) |^ 3)) or x1 = 3-root(-q/2-sqrt(q^2/4+(p/3) |^ 3)) + 3-root(-q/2-sqrt(q^2/4+
  (p/3) |^ 3)) by A10,A11,POLYEQ_1:19;
  hence thesis by A4,A10,A8,A9,COMPLEX1:13;
end;
