
theorem Th1:
for x,y being Complex st Im(x) = 0 & Re(y) = 0 holds Re(x/y) = 0
proof
let x,y be Complex;
reconsider R2 = (Re y)^2, I2 = (Im y)^2 as Real;
assume A1: Im(x) = 0 & Re(y) = 0;
then Re(x/y)
   = (Re x * 0 + Im x * Im y) / (R2 + I2) by COMPLEX1:24
  .= 0 / (R2 + I2) by A1;
hence Re(x/y) = 0;
end;
