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
  z^2 = s & Im s = 0 & Re s < 0 implies z= sqrt (-Re s)*<i> or z= -sqrt
  (-Re s)*<i>
proof
  assume
A1: z^2=s;
  assume that
A2: Im s =0 and
A3: Re s < 0;
  z=sqrt (( Re s+sqrt ((Re s)^2+0))/2)+ (sqrt ((-Re s+sqrt ((Re s)^2+0))/2
))*<i> or z=-sqrt (( Re s+sqrt ((Re s)^2+0))/2)+ (-sqrt ((-Re s+sqrt ((Re s)^2+
  0^2))/2))*<i> by A1,A2,Th19;
  then
  z=sqrt (( Re s+-Re s)/2)+ sqrt ((-Re s+sqrt ((Re s)^2+0))/2)*<i> or z=-
  sqrt (( Re s+sqrt ((Re s)^2+0))/2)+ (-sqrt ((-Re s+sqrt ((Re s)^2+0))/2))*<i>
  by A3,SQUARE_1:23;
  then
  z=sqrt (( Re s+-Re s)/2)+ sqrt ((-Re s+-Re s)/2)*<i> or z=-sqrt (( Re s+
  -Re s)/2)+(-sqrt ((-Re s+sqrt ((Re s)^2+0))/2))*<i> by A3,SQUARE_1:23;
  then
  z=sqrt (( Re s+-Re s)/2)+ sqrt ((-Re s+-Re s)/2)*<i> or z=-sqrt (( Re s+
  -Re s)/2)+(-sqrt ((-Re s+-Re s)/2))*<i> by A3,SQUARE_1:23;
  hence thesis;
end;
