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 Th73:
  for H,N be normal Subgroup of G
   ex M being strict normal Subgroup of G st the carrier of M = N ~ H
proof
  let H,N be normal Subgroup of G;
  consider M being strict Subgroup of G such that
A1: the carrier of M = N ~ H by Th71;
  for x be Element of G holds x * M c= M * x
  proof
    let x be Element of G;
    let y be object;
    assume
A2: y in x * M;
    then reconsider y as Element of G;
    consider z be Element of G such that
A3: y = x * z & z in M by A2,GROUP_2:103;
     z in the carrier of M by A3,STRUCT_0:def 5;
    then consider z9 be Element of G such that
A4: z9 in z * N & z9 in H by Th3,A1,Th51;
    x * z9 * x" in H by A4,Th4;
    then
A5: x * z9 * x" in carr(H) by STRUCT_0:def 5;
A6: x * z9 * x" in x * (z * N) * x" by A4,GROUP_8:15;
    x * (z * N) * x" = x * ((z * N) * x") by GROUP_2:33
                    .= x * (z * (N * x")) by GROUP_2:33
                    .= x * (z * (x" * N)) by GROUP_3:117
                    .= x * (z * x" * N) by GROUP_2:32
                    .= x * (z * x") * N by GROUP_2:32
                    .= (x * z * x") * N by GROUP_1:def 3;
     then (x * z * x") * N meets carr(H) by A5,A6,XBOOLE_0:3;
     then x * z * x" in N ~ H;
     then
A7:  x * z * x" in M by A1,STRUCT_0:def 5;
     (x * z * x") * x = y by A3,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 A1;
end;
