reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th50:
  for a being Complex holds Rotate(a, r) = 0 iff a = 0
proof
  let a be Complex;
  hereby
    assume Rotate(a, r) = 0;
    then
A1: Rotate(a, r) = 0+0*<i>;
    per cases by A1,COMPLEX1:77;
    suppose
      |.a.| = 0;
      hence a = 0 by COMPLEX1:45;
    end;
    suppose
      cos (r+Arg a) = 0 & sin (r+Arg a) = 0;
      hence a = 0 by Th10;
    end;
  end;
  assume a = 0;
  hence thesis by COMPLEX1:44;
end;
