
theorem Th28:
  for i,j being object, A, B, C, D being set st A c= C & B c= D
  holds product((i,j) --> (A,B)) c= product((i,j) --> (C,D))
proof
  let i,j be object, A,B,C,D be set;
  assume A1: A c= C & B c= D;
  per cases;
  suppose A2: i <> j;
      let x be object;
      assume x in product((i,j) --> (A,B));
      then consider g being Function such that
        A3: g = x & dom g = dom (i,j) --> (A,B) and
        A4: for y being object st y in dom (i,j) --> (A,B)
          holds g.y in ((i,j) --> (A,B)).y by CARD_3:def 5;
      A5: dom (i,j) --> (A,B) = {i,j} by FUNCT_4:62
        .= dom (i,j) --> (C,D) by FUNCT_4:62;
      for y being object st y in dom (i,j) --> (C,D) holds
        g.y in ((i,j) --> (C,D)).y
      proof
        let y be object;
        assume A6: y in dom (i,j) --> (C,D);
        then y in {i,j} by FUNCT_4:62;
        then per cases by TARSKI:def 2;
        suppose A7: y = i;
          then g.y in ((i,j) --> (A,B)).i by A4, A5, A6;
          then g.y in A by A2, FUNCT_4:63;
          then g.y in C by A1;
          hence thesis by A2, A7, FUNCT_4:63;
        end;
        suppose A8: y = j;
          then g.y in ((i,j) --> (A,B)).j by A4, A5, A6;
          then g.y in B by FUNCT_4:63;
          then g.y in D by A1;
          hence thesis by A8, FUNCT_4:63;
        end;
      end;
      hence thesis by A3, A5, CARD_3:def 5;
  end;
  suppose A9: i = j;
    then A10: (i,j) --> (A,B) = i .--> B by FUNCT_4:81
      .= {i} --> B by FUNCOP_1:def 9;
    (i,j) --> (C,D) = i .--> D by A9, FUNCT_4:81
      .= {i} --> D by FUNCOP_1:def 9;
    hence thesis by A1, A10, Th14;
  end;
end;
