reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem Th66:
  for G being finite Group
  for H1,H2 being strict Subgroup of G
  st card(H1 /\ H2) = card H1 & card(H1 /\ H2) = card H2
  holds H1 = H2
proof
  let G be finite Group;
  let H1,H2 be strict Subgroup of G;
  assume A1: card(H1 /\ H2) = card H1;
  assume A2: card(H1 /\ H2) = card H2;
  A3: H1 /\ H2 = H1
  proof
    reconsider H12 = H1 /\ H2 as strict Subgroup of H1 by GROUP_2:88;
    multMagma(# the carrier of H12, the multF of H12 #)
    = multMagma(# the carrier of H1, the multF of H1 #) by A1,GROUP_2:73;
    hence thesis;
  end;
  H1 /\ H2 = H2
  proof
    reconsider H12 = H1 /\ H2 as strict Subgroup of H2 by GROUP_2:88;
    multMagma(# the carrier of H12, the multF of H12 #)
    = multMagma(# the carrier of H2, the multF of H2 #) by A2,GROUP_2:73;
    hence thesis;
  end;
  hence thesis by A3;
end;
