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 Th23:
  z^2 = s implies z=sqrt (( Re s+sqrt ((Re s)^2+(Im s)^2))/2)+ (
sqrt ((-Re s+sqrt ((Re s)^2+(Im s)^2))/2))*<i> or z=-sqrt (( Re s+sqrt ((Re s)
^2+(Im s)^2))/2)+ (-sqrt ((-Re s+sqrt ((Re s)^2+(Im s)^2))/2))*<i> or z= sqrt (
( Re s+sqrt ((Re s)^2+(Im s)^2))/2)+ (-sqrt ((-Re s+sqrt ((Re s)^2+(Im s)^2))/2
))*<i> or z=-sqrt (( Re s+sqrt ((Re s)^2+(Im s)^2))/2)+ (sqrt ((-Re s+sqrt ((Re
  s)^2+(Im s)^2))/2))*<i>
proof
  set a=Re s;
  set b=Im s;
  set u=Re z;
  set v=Im z;
A1: u^2>=0 & v^2>=0 by XREAL_1:63;
  z = u + v*<i> by COMPLEX1:13;
  then
A2: s=a+b*<i> & z^2 = u^2-v^2 + 2*u*v*<i> by COMPLEX1:13;
  assume
A3: z^2=s;
  then
A4: u^2-v^2=a by A2,COMPLEX1:77;
  then sqrt (u^2+v^2)^2=sqrt (a^2+b^2) by A3,A2;
  then
A5: u^2+v^2=sqrt (a^2+b^2) by A1,SQUARE_1:22;
  per cases;
  suppose
    b>=0;
    then u<=0 & v<=0 or u>=0 & v>=0 by A3,A2,COMPLEX1:77;
    then -u=sqrt ((a+sqrt (a^2+b^2))/2) & -v=sqrt ((-a+sqrt (a^2+b^2))/ 2) or
    u=sqrt ((a+sqrt (a^2+b^2))/2) & v=sqrt ((-a+sqrt (a^2+b^2))/2) by A4,A5,
SQUARE_1:22,23;
    hence thesis by COMPLEX1:13;
  end;
  suppose
    b<0;
    then 2*(u*v)<0 by A3,A2,COMPLEX1:77;
    then u*v<0;
    then u<0 & v>0 or u>0 & v<0 by XREAL_1:133;
    then
    -u=sqrt ((a+sqrt (a^2+b^2))/2) & v=sqrt ((-a+sqrt (a^2+b^2))/2) or u=
    sqrt ((a+sqrt (a^2+b^2))/2) & -v=sqrt ((-a+sqrt (a^2+b^2))/2) by A4,A5,
SQUARE_1:22,23;
    hence thesis by COMPLEX1:13;
  end;
end;
