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 Th55:
  for G being Group
  for H1,H2 being Subgroup of G
  st H1 is Subgroup of H2
  for a being Element of G
  holds H1 |^ a is Subgroup of H2 |^ a
proof
  let G be Group;
  let H1,H2 be Subgroup of G;
  assume A1: H1 is Subgroup of H2;
  let a be Element of G;
  for h being Element of G st h in H1 |^ a holds h in H2 |^ a
  proof
    let h be Element of G;
    assume h in H1 |^ a;
    then consider g being Element of G such that
    B1: h = g |^ a & g in H1
    by GROUP_3:58;
    g in H2 by A1,B1,GROUP_2:40;
    hence thesis by B1,GROUP_3:58;
  end;
  hence H1 |^ a is Subgroup of H2 |^ a by GROUP_2:58;
end;
