
theorem ProdMon:
  for x, y, z, w being Real st
    |. x - y .| < |. z - w .| holds
      (x - y) ^2 < (z - w) ^2
  proof
    let x, y, z, w be Real;
A2: |. x - y .| ^2 = (x - y) ^2 by COMPLEX1:75;
    assume |. x - y .| < |. z - w .|; then
    |. x - y .| ^2 < |. z - w .| ^2 by SQUARE_1:16;
    hence thesis by COMPLEX1:75,A2;
  end;
