 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 ThCarrG:
  for G being Group
  for I being non empty set
  for F being componentwise_strict Subgroup-Family of I,G
  for Fam being Subset of Subgroups G st Fam = rng F
  holds union { A where A is Subset of G : ex H being strict Subgroup of G
  st H in Fam & A = the carrier of H } = Union (Carrier F)
proof
  let G be Group;
  let I be non empty set;
  let F be componentwise_strict Subgroup-Family of I,G;
  let Fam be Subset of Subgroups G;
  assume A1: Fam = rng F;
  set X = { A where A is Subset of G : ex H being strict Subgroup of G
                                       st H in Fam & A = the carrier of H };
  for x being object holds x in X iff x in rng (Carrier F)
  proof
    let x be object;
    hereby
      assume x in X;
      then consider A being Subset of G such that
      A2: x = A &
      ex H being strict Subgroup of G st H in Fam & A = the carrier of H;
      consider H being strict Subgroup of G such that
      A3: H in Fam & x = the carrier of H by A2;
      consider i being Element of I such that
      A4: H = F.i by A1, A3, MssRng;
      x = (Carrier F).i by A3,A4,Th9;
      hence x in rng (Carrier F) by MssRng;
    end;
    assume x in rng (Carrier F);
    then consider i being Element of I such that
    A5: x = (Carrier F).i by MssRng;
    F.i is strict Subgroup of G by Def19;
    then consider H being strict Subgroup of G such that
    A6: H = F.i;
    A8: H in Fam by A1, A6, MssRng;
    ex A being Subset of G st x = A & ex H0 being strict Subgroup
    of G st A = the carrier of H0 & H0 in Fam
    proof
      take A = carr H;
      thus thesis by A8, A5, A6, Th9;
    end;
    hence x in X;
  end;
  then X = rng (Carrier F) by TARSKI:2;
  hence union X = Union (Carrier F) by CARD_3:def 4;
end;
