
theorem Th14:
  for X being non empty set, Y being set, Z being Subset of Y
  holds product(X --> Z) c= product (X --> Y)
proof
  let X be non empty set, Y be set, Z be Subset of Y;
    let x be object;
    assume x in product(X --> Z);
    then consider g being Function such that
      A1: x = g & dom g = dom (X --> Z) and
      A2: for y being object st y in dom (X --> Z) holds g.y in (X --> Z).y
      by CARD_3:def 5;
    now
      let y be object;
      assume a4: y in dom (X --> Y);
      then A4: y in X;
      A5: (X --> Z).y = Z & (X --> Y).y = Y by a4,FUNCOP_1:7;
      y in dom (X --> Z) by A4;
      then g.y in (X --> Z).y by A2;
      hence g.y in (X --> Y).y  by A5;
    end;
    hence thesis by A1, CARD_3:def 5;
end;
