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 Th5:
  Re z^3 = (Re z)|^3 - 3*Re z*(Im z)^2 & Im z^3 = -(Im z)|^ 3+3*( Re z)^2*Im z
proof
  set x = Re z;
  set y = Im z;
  Re z + (Im z)*<i> = z by COMPLEX1:13;
  then
  z^3 = Re(x^2-y^2+(2*(x*y))*<i>) *Re (x+y*<i>)-Im(x^2-y^2+(2*(x*y))*<i>)
*Im (x+y*<i>)+(Re(x^2-y^2+(2*(x*y))*<i>) *Im (x+y*<i>)+ Re (x+y*<i>)*Im (x^2-y
  ^2+(2*(x*y))*<i>))*<i> by COMPLEX1:82
    .= ((x^2-y^2) *Re (x+y*<i>)-Im(x^2-y^2+(2*(x*y))*<i>) *Im (x+y*<i>))+(Re
(x^2-y^2+(2*(x*y))*<i>) *Im (x+y*<i>)+ Re (x+y*<i>)*Im (x^2-y^2+(2*(x*y))*<i>))
  *<i> by COMPLEX1:12
    .= ((x^2-y^2) *Re (x+y*<i>)-(2*(x*y)) *Im (x+y*<i>))+(Re(x^2-y^2+(2*(x*y
))*<i>) *Im (x+y*<i>)+ Re (x+y*<i>)*Im (x^2-y^2+(2*(x*y))*<i>))*<i> by
COMPLEX1:12
    .= ((x^2-y^2) *Re (x+y*<i>)-(2*(x*y)) *Im (x+y*<i>))+((x^2-y^2) *Im (x+y
  *<i>)+ Re (x+y*<i>)*Im (x^2-y^2+(2*(x*y))*<i>))*<i> by COMPLEX1:12
    .= ((x^2-y^2) *Re (x+y*<i>)-(2*(x*y))*Im (x+y*<i>) )+((x^2-y^2) *Im (x+y
  *<i>)+Re (x+y*<i>)*(2*(x*y)))*<i> by COMPLEX1:12
    .= ((x^2-y^2) *x-(2*(x*y))*Im (x+y*<i>) )+((x^2-y^2) *Im (x+y*<i>)+Re (x
  +y*<i>)*(2*(x*y)))*<i> by COMPLEX1:12
    .= ((x^2-y^2) *x-(2*(x*y))*y )+((x^2-y^2) *Im (x+y*<i>)+Re (x+y*<i>)*(2*
  (x*y)))*<i> by COMPLEX1:12
    .= ((x^2-y^2) *x-(2*(x*y))*y )+((x^2-y^2) *y+Re (x+y*<i>)*(2*(x*y)))*<i>
  by COMPLEX1:12
    .= ((x^2*x-y^2*x+0*x)-2*(x*y)*y)+((x^2-y^2+0)*y+x*(2*(x*y)))*<i> by
COMPLEX1:12
    .= ((x|^1*x*x-y^2*x)-2*(x*y)*y)+((x^2*y-y^2*y)+x*(2*(x*y)))*<i>
    .= ((x|^ (1+1)*x-y^2*x)-2*(x*y)*y)+((x^2*y-y^2*y)+x*(2*(x*y)))*<i> by
NEWTON:6
    .= ((x|^(2+1)-y^2*x)-2*(x*y)*y)+((x^2*y-y^2*y)+x*(2*(x*y)))*<i> by NEWTON:6
    .= ((x|^ 3-y^2*x)-2*(x*y)*y)+((x^2*y-y|^ 1*y*y)+x*(2*(x*y)))*<i>
    .= ((x|^ 3-y^2*x)-2*(x*y)*y)+((x^2*y-y|^ (1+1)*y)+x*(2*(x*y)))*<i> by
NEWTON:6
    .= ((x|^ 3-y^2*x)-2*(x*(y*y)))+((x^2*y-y|^ (2+1))+x*(2*(x*y)))*<i> by
NEWTON:6
    .= (x|^ 3-3*x*y^2 )+(-y|^ 3+3*x^2*y)*<i>;
  hence thesis by COMPLEX1:12;
end;
