theorem Th80:
  (for H being Subgroup of G st H = H1 /\ H2 holds the carrier of
  H = (the carrier of H1) /\ (the carrier of H2)) & for H being strict Subgroup
  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 Subgroup of G st H = H1 /\ H2 holds the carrier of H = (the
  carrier of H1) /\ (the carrier of H2)
  proof
    let H be Subgroup of G;
    assume H = H1 /\ H2;
    hence the carrier of H = carr(H1) /\ carr(H2) by Def10
      .= (the carrier of H1) /\ (the carrier of H2);
  end;
  let H be strict Subgroup of G;
  assume the carrier of H = (the carrier of H1) /\ (the carrier of H2);
  hence thesis by A1,Def10;
end;
