 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
  for F1,F2 being componentwise_strict Subgroup-Family of F
  st (for i being Element of I
      holds Image (proj (F1, i)) is Subgroup of Image (proj (F2, i)))
  holds (product F1) is strict Subgroup of (product F2)
proof
  let F1,F2 be componentwise_strict Subgroup-Family of F;
  assume A1: for i being Element of I
  holds Image (proj (F1, i)) is Subgroup of Image (proj (F2, i));
  for i being Element of I holds F1.i is Subgroup of F2.i
  proof
    let i be Element of I;
    Image (proj (F1, i)) is Subgroup of Image (proj (F2, i)) by A1;
    hence F1.i is Subgroup of F2.i by Th50;
  end;
  hence thesis by Th49;
end;
