
theorem
  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 H1 is Subgroup of H2
proof
  let G be Group;
  let H1, H2 be Subgroup of G;
  let p, q be Element of lattice G;
  assume that
A1: p = H1 and
A2: q = H2;
  thus p [= q implies H1 is Subgroup of H2
  proof
    assume p [= q;
    then the carrier of H1 c= the carrier of H2 by A1,A2,Th25;
    hence thesis by GROUP_2:57;
  end;
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 H1 is Subgroup of H2 implies p [= q
  proof
    assume H1 is Subgroup of H2;
    then
A5: 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,A5,LATTICES:4;
  end;
end;
