 reserve I for non empty set;
 reserve i for Element of I;
 reserve F for Group-Family of I;
 reserve G for Group;
reserve S for Subgroup-Family of F;
reserve f for Homomorphism-Family of G, F;

theorem :: missing
  for A being ManySortedSet of I
  st (for i being Element of I holds A.i is Subset of F.i)
  holds product A is Subset of product F
proof
  let A be ManySortedSet of I;
  assume A1: for i being Element of I holds A.i is Subset of F.i;
  for x being object st x in product A holds x in the carrier of product F
  proof
    let x be object;
    assume B1: x in product A;
    reconsider x as Function by B1;
    B2: dom x = dom A by B1, CARD_3:9;
    B3: dom F = I & dom A = I by PARTFUN1:def 2;
    for i being Element of I holds x.i in F.i
    proof
      let i be Element of I;
      C1: A.i is Subset of F.i by A1;
      x.i in A.i by B1, B3, CARD_3:9;
      hence x.i in F.i by C1;
    end;
    then x in product F by B2, Th47, PARTFUN1:def 2;
    hence thesis;
  end;
  hence product A is Subset of product F by TARSKI:def 3;
end;
