reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;
reserve N1,N2 for Subgroup of G;

theorem Th74:
  for H,N being normal Subgroup of G st N is Subgroup of H
  ex M being strict normal Subgroup of G st the carrier of M = N ` H
proof
  let H,N be normal Subgroup of G;
  assume
A1: N is Subgroup of H;
  then consider M being strict Subgroup of G such that
A2: the carrier of M = N ` H by Th72;
  for x be Element of G holds x * M c= M * x
  proof
    let x be Element of G;
    let y be object;
    assume
A3: y in x * M;
    then reconsider y as Element of G;
    consider z be Element of G such that
A4: y = x * z & z in M by A3,GROUP_2:103;
    z in the carrier of M by A4,STRUCT_0:def 5;
    then
A5: z * N c= carr(H) by A2,Th49;
    z in z * N by GROUP_2:108;
    then z in H by A5,STRUCT_0:def 5;
    then x * z * x" in H by Th4;
    then
A6: (x * z * x") * H = carr(H) by GROUP_2:113;
    (x * z * x") * N c= (x * z * x") * H by A1,GROUP_3:6;
    then x * z * x" in N ` H by A6; then
A7: x * z * x" in M by A2,STRUCT_0:def 5;
    (x * z * x") * x = y by A4,GROUP_3:1;
    hence thesis by A7,GROUP_2:104;
  end;
  then M is normal Subgroup of G by GROUP_3:118;
  hence thesis by A2;
end;
