reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;

theorem Th19:
  for A, B, C, D being set, a, b being object st A c= B & C c= D
  holds product (a,b) --> (A,C) c= product (a,b) --> (B,D)
proof
  let A, B, C, D being set, a, b be object such that
A1: A c= B and
A2: C c= D;
  set f = (a,b) --> (A,C), g = (a,b) --> (B,D);
A3: dom f = {a,b} by FUNCT_4:62;
A4: for x being object st x in dom f holds f.x c= g.x
  proof
    let x be object;
    assume x in dom f;
    then
A5: x = a or x = b by A3,TARSKI:def 2;
    per cases;
    suppose
A6:   a <> b;
A7:   f.b = C by FUNCT_4:63;
      f.a = A by A6,FUNCT_4:63;
      hence thesis by A1,A2,A5,A6,A7,FUNCT_4:63;
    end;
    suppose
A8:   a = b;
      then f = a .--> C by FUNCT_4:81;
      then
A9:   f.a = C by FUNCOP_1:72;
      g = a .--> D by A8,FUNCT_4:81;
      hence thesis by A2,A5,A8,A9,FUNCOP_1:72;
    end;
  end;
  dom g = {a,b} by FUNCT_4:62;
  hence thesis by A4,CARD_3:27,FUNCT_4:62;
end;
