reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th16:
  for I being set
  for A,B being ManySortedSet of I st A c= B
  holds product A c= product B
  proof
    let I be set;
    let A,B be ManySortedSet of I;
    assume A1: A c= B;
    let x be object; assume x in product A; then
    consider g being Function such that
A2: x = g & dom g = dom A & for y being object st y in dom A holds g.y in A.y
    by CARD_3:def 5;
A3: dom A = I & dom B = I by PARTFUN1:def 2;
    now let y be object; assume y in I; then
      g.y in A.y & A.y c= B.y by A1,A2,A3;
      hence g.y in B.y;
    end;
    hence x in product B by A2,A3,CARD_3:9;
  end;
