
theorem Th25:
  for G being Group for H1, H2 being Subgroup of G for p, q being
  Element of lattice G st p = H1 & q = H2 holds p [= q iff the carrier of H1 c=
  the carrier of H2
proof
  let G be Group;
  let H1, H2 be Subgroup of G;
  let p, q be Element of lattice G such that
A1: p = H1 and
A2: q = H2;
A3: H1 is strict Subgroup of G by A1,GROUP_3:def 1;
A4: H2 is strict Subgroup of G by A2,GROUP_3:def 1;
  thus p [= q implies the carrier of H1 c= the carrier of H2
  proof
    assume p [= q;
    then
A5: p "/\" q = p by LATTICES:4;
    p "/\" q = (the L_meet of lattice G).(p,q) by LATTICES:def 2
      .= H1 /\ H2 by A1,A2,A3,A4,GROUP_4:def 11;
    then
    (the carrier of H1) /\ the carrier of H2 = the carrier of H1 by A1,A5,Th1;
    hence thesis by XBOOLE_1:17;
  end;
  thus the carrier of H1 c= the carrier of H2 implies p [= q
  proof
    assume the carrier of H1 c= the carrier of H2;
    then H1 is Subgroup of H2 by GROUP_2:57;
    then
A6: H1 /\ H2 = H1 by A3,GROUP_2:89;
    H1 /\ H2 = (the L_meet of lattice G).(p,q) by A1,A2,A3,A4,GROUP_4:def 11
      .= p "/\" q by LATTICES:def 2;
    hence thesis by A1,A6,LATTICES:4;
  end;
end;
