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 Th26:
  z1<>0 & Polynom(z1,z2,z3,z)=0 implies for h,t st h=(z2/(2*z1))^2
-z3/z1 & t=z2/(2*z1) holds z= sqrt (( Re h+sqrt ((Re h)^2+(Im h)^2))/2)+ (sqrt
((-Re h+sqrt ((Re h)^2+(Im h)^2))/2))*<i>-t or z=-sqrt (( Re h+sqrt ((Re h)^2+(
Im h)^2))/2)+ (-sqrt ((-Re h+sqrt ((Re h)^2+(Im h)^2))/2))*<i>-t or z= sqrt ((
Re h+sqrt ((Re h)^2+(Im h)^2))/2)+ (-sqrt ((-Re h+sqrt ((Re h)^2+(Im h)^2))/2))
*<i>-t or z=-sqrt (( Re h+sqrt ((Re h)^2+(Im h)^2))/2)+ (sqrt ((-Re h+sqrt ((Re
  h)^2+(Im h)^2))/2))*<i>-t
proof
  assume that
A1: z1<>0 and
A2: Polynom(z1,z2,z3,z)=0;
  (z1*z^2+z2*z+z3)/z1=0 by A2;
  then (z^2*z1)/z1+(z2*z)/z1+z3/z1=0;
  then z^2+(z2/z1)*z+z3/z1=0 by A1,XCMPLX_1:89;
  then z^2+2*z2/(2*z1)*z+z3/z1=0 by XCMPLX_1:91;
  then
A3: (z+(z2/(2*z1)))^2-((z2/(2*z1))^2-z3/z1)=0;
  let h,t;
  assume h=(z2/(2*z1))^2-z3/z1 & t=z2/(2*z1);
  then
  z+t-t= sqrt (( Re h+sqrt ((Re h)^2+(Im h)^2))/2)+ (sqrt ((-Re h+sqrt ((
Re h)^2+(Im h)^2))/2))*<i>-t or z+t-t=-sqrt (( Re h+sqrt ((Re h)^2+(Im h)^2))/2
  )+ (-sqrt ((-Re h+sqrt ((Re h)^2+(Im h)^2))/2))*<i>-t or z+t-t= sqrt (( Re h+
sqrt ((Re h)^2+(Im h)^2))/2)+ (-sqrt ((-Re h+sqrt ((Re h)^2+(Im h)^2))/2))*<i>-
t or z+t-t=-sqrt (( Re h+sqrt ((Re h)^2+(Im h)^2))/2)+ (sqrt ((-Re h+sqrt ((Re
  h)^2+(Im h)^2))/2))*<i>-t by A3,Th23;
  hence thesis;
end;
