reserve x,y for set,
  G for Group,
  A,B,H,H1,H2 for Subgroup of G,
  a,b,c for Element of G,
  F,F1 for FinSequence of the carrier of G,
  I,I1 for FinSequence of INT,
  i,j for Element of NAT;

theorem Th15:
  A is normal Subgroup of G iff [.A,(Omega).G.] is Subgroup of A
proof
  thus A is normal Subgroup of G implies [.A,(Omega).G.] is
  Subgroup of A by Lm3;
  assume
A1: [.A,(Omega).G.] is Subgroup of A;
  for b holds b * A c= A * b
  proof
    let b;
    let x be object;
    assume
A2: x in b * A;
    then reconsider x as Element of G;
    consider Z be Element of G such that
A3: x = b * Z & Z in A by A2,GROUP_2:103;
A4: Z" in A by A3,GROUP_2:51;
    b" in (Omega).G by STRUCT_0:def 5; then
 [.b",Z".] in [.(Omega).G,A.] by A4,GROUP_5:65;
    then [.b",Z".] in A by A1,GROUP_2:40; then
A5: (b * Z * b" * Z") * Z in A by A3,GROUP_2:50;
A6: (b * Z * b" * Z") * Z = ((b * Z) * b") * (Z" * Z) by GROUP_1:def 3
                         .= ((b * Z) * b") * 1_G by GROUP_1:def 5
                         .= b * Z * b" by GROUP_1:def 4;
    (b * Z * b") * b = (b * Z) * (b" * b) by GROUP_1:def 3
                    .= b * Z * 1_G by GROUP_1:def 5
                    .= b * Z  by GROUP_1:def 4;
    hence thesis by A3,A5,A6,GROUP_2:104;
  end;
  hence thesis by GROUP_3:118;
end;
