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
  (-a+sqrt (a^2+b^2))/2 >=0 & (a+sqrt (a^2+b^2))/2>=0
proof
A1: b^2>=0 by XREAL_1:63;
A2: a^2>=0 by XREAL_1:63;
  then
A3: sqrt (a^2+b^2)>=0 by A1,SQUARE_1:def 2;
  a^2+b^2>=0+a^2 by A1,XREAL_1:7;
  then
A4: sqrt(a^2+b^2)>=sqrt(a^2) by A2,SQUARE_1:26;
  per cases;
  suppose
A5: a>=0;
    then sqrt(a^2+b^2)>=a by A4,SQUARE_1:22;
    then sqrt(a^2+b^2)-a>=a-a by XREAL_1:9;
    hence thesis by A3,A5;
  end;
  suppose
A6: a<0;
    then sqrt(a^2+b^2) >= -a by A4,SQUARE_1:23;
    then sqrt(a^2+b^2)-(-a) >= -a-(-a) by XREAL_1:9;
    hence thesis by A3,A6;
  end;
end;
