 reserve G, A for Group;
 reserve phi for Homomorphism of A,AutGroup(G);

theorem Th8:
  for G being Group
  for H,K being strict Subgroup of G
  st H <> K & K is Subgroup of H
  ex g being Element of G
  st g in H & not g in K
proof
  let G be Group;
  let H,K be strict Subgroup of G;
  assume A1: H <> K;
  assume A2: K is Subgroup of H;
  then not H is Subgroup of K by A1, GROUP_2:55;
  then the carrier of K c= the carrier of H
  & the carrier of H <> the carrier of K
    by A2, GROUP_2:def 5, GROUP_2:57;
  then consider x being object such that
  A3: x in the carrier of H & not x in the carrier of K
    by XBOOLE_0:6, XBOOLE_0:def 8;
  x in H & not x in K by A3;
  then x in G by GROUP_2:40;
  then reconsider g = x as Element of G;
  take g;
  thus g in H & not g in K by A3;
end;
