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 Th19:
  z^2 = s & Im s >= 0 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>
proof
  set a=Re s;
  set b=Im s;
  set u=Re z;
  set v=Im z;
A1: z = u+v*<i> by COMPLEX1:13;
  assume z^2=s;
  then
A2: u^2-v^2+2*u*v*<i> =a+b*<i> by A1,COMPLEX1:13;
  assume Im s >=0;
  then
A3: u<=0 & v<=0 or u>=0 & v>=0 by A2,COMPLEX1:77;
A4: u^2>=0 & v^2>=0 by XREAL_1:63;
A5: u^2-v^2=a by A2,COMPLEX1:77;
  then (u^2+v^2)^2=a^2+b^2 by A2;
  then u^2+v^2=sqrt (a^2+b^2) by A4,SQUARE_1:22;
  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 A5,A3,
SQUARE_1:22,23;
  hence thesis by COMPLEX1:13;
end;
