
theorem
  for x,y being Real st x * y < 0 holds 0 < x / (x - y) < 1
  proof
    let x,y be Real;
    assume
A1: x * y < 0;
    then x <> 0 & y <> 0;
    then per cases;
    suppose
A2:   x < 0;
      then y > 0 by A1;
      then x - y < x by XREAL_1:44;
      hence thesis by A2,XREAL_1:190;
    end;
    suppose
A3:   0 < x;
      then y < 0 by A1;
      then x < x - y by XREAL_1:46;
      hence thesis by A3,XREAL_1:189;
    end;
  end;
