reserve i for Integer,
  a, b, r, s for Real;

theorem Th3:
  for x, y, X, Y being set, f being Function st x <> y & f in
  product ((x,y) --> (X,Y)) holds f.x in X & f.y in Y
proof
  let x, y, X, Y be set, f be Function such that
A1: x <> y and
A2: f in product ((x,y) --> (X,Y));
  set g = (x,y) --> (X,Y);
A3: dom g = {x,y} by FUNCT_4:62;
  then y in dom g by TARSKI:def 2;
  then
A4: f.y in g.y by A2,CARD_3:9;
  x in dom g by A3,TARSKI:def 2;
  then f.x in g.x by A2,CARD_3:9;
  hence thesis by A1,A4,FUNCT_4:63;
end;
