 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 CSSubFam:
  for F being componentwise_strict Subgroup-Family of I, G
  holds F is (Subgroups G)-valued
proof
  let F be componentwise_strict Subgroup-Family of I, G;
  for y being object st y in rng F holds y in Subgroups G
  proof
    let y be object;
    assume y in rng F;
    then consider i being Element of I such that
    A1: y = F.i by MssRng;
    y is strict Subgroup of G by A1, ThS2;
    hence y in Subgroups G by GROUP_3:def 1;
  end;
  hence F is (Subgroups G)-valued by RELAT_1:def 19, TARSKI:def 3;
end;
