
theorem Th22:
  for I be non empty set,
      J be non-empty ManySortedSet of I,
      F be Group-Family of I,J
  holds dsum F is Subgroup of dprod F
  proof
    let I be non empty set,
        J be non-empty ManySortedSet of I,
        F be Group-Family of I,J;
    for i be Element of I
    holds (sum_bundle F).i is Subgroup of (prod_bundle F).i
    proof
      let i be Element of I;
      (sum_bundle F).i = sum F.i by Def7;
      hence thesis by Def6;
    end; then
    product(sum_bundle F) is Subgroup of product(prod_bundle F) by Th20;
    hence thesis by GROUP_2:56;
  end;
