reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;

theorem Th22:
  union Cosets N = the carrier of G
proof
  reconsider H = the multMagma of N as strict normal Subgroup of G by Lm6;
  now
    set h = the Element of H;
    let x be object;
    reconsider g = h as Element of G by GROUP_2:42;
    assume x in the carrier of G;
    then reconsider a = x as Element of G;
A1: a = a * 1_G by GROUP_1:def 4
      .= a * (g" * g) by GROUP_1:def 5
      .= a * g" * g by GROUP_1:def 3;
A2: a * g" * H in Cosets H by GROUP_2:def 15;
    h in H by STRUCT_0:def 5;
    then a in a * g" * H by A1,GROUP_2:103;
    hence x in union Cosets H by A2,TARSKI:def 4;
  end;
  then
A3: the carrier of G c= union Cosets H;
  Cosets N = Cosets H by Def14;
  hence thesis by A3,XBOOLE_0:def 10;
end;
