reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;

theorem Th18:
  (for H being StableSubgroup of G st H = H1 /\ H2 holds the
carrier of H = (the carrier of H1) /\ (the carrier of H2)) & for H being strict
  StableSubgroup of G holds the carrier of H = (the carrier of H1) /\ (the
  carrier of H2) implies H = H1 /\ H2
proof
A1: the carrier of H1 = carr(H1) & the carrier of H2 = carr(H2);
  thus for H being StableSubgroup of G st H = H1 /\ H2 holds the carrier of H
  = (the carrier of H1) /\ (the carrier of H2)
  proof
    let H be StableSubgroup of G;
    assume H = H1 /\ H2;
    hence the carrier of H = carr(H1)/\carr(H2) by Def25
      .= (the carrier of H1)/\(the carrier of H2);
  end;
  let H be strict StableSubgroup of G;
  assume the carrier of H = (the carrier of H1) /\ (the carrier of H2);
  hence thesis by A1,Def25;
end;
