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 Th19:
  for G1 being Subgroup of G,N being strict normal Subgroup of G st
  N is Subgroup of G1 & G1./.(G1,N)`*` is Subgroup of center (G./.N) holds
   [.G1,(Omega).G.] is Subgroup of N
proof
  let G1 be Subgroup of G;
  let N be strict normal Subgroup of G;
  assume that
A1: N is Subgroup of G1 and
A2: G1./.(G1,N)`*` is Subgroup of center (G./.N);
A3: (G1,N)`*` = N by A1,GROUP_6:def 1;
  reconsider J = G1./.(G1,N)`*` as Subgroup of G./.N by A1,GROUP_6:28;
  reconsider I = N as normal Subgroup of G1 by A3;
A4: commutators (G1,(Omega).G) c= carr N
  proof
    now
      let x be Element of G;
      assume x in commutators (G1,(Omega).G);
      then consider a,b such that
A5:   x = [.a,b.] & a in G1 & b in (Omega).G by GROUP_5:52;
      reconsider c = a as Element of G1 by A5,STRUCT_0:def 5;
      reconsider S9 = c * I as Element of J by A3,GROUP_6:22;
A6:   S9 in J by STRUCT_0:def 5;
      reconsider T = b * N  as Element of G./.N
      by GROUP_6:14;
      reconsider d = c as Element of G;
   d * N = c * I by GROUP_6:2;
      then reconsider S = S9 as Element of G./.N by GROUP_6:14;
      S in center (G./.N) by A2,A6,GROUP_2:40; then
A7:   S * T = T * S by GROUP_5:77;
A8:   S = d * N & T = b * N & @S = S & @T = T by GROUP_6:2;
then A9:  S * T = (d * N) * (b * N) by GROUP_6:def 3
           .= d * (N * (b * N)) by GROUP_3:10
           .= d * (N * b * N) by GROUP_3:13
           .= d * (b * N * N) by GROUP_3:117
           .= d * (b * N) by GROUP_6:5
           .= d * b * N by GROUP_2:105;
      T * S = (b * N) * (d * N) by A8,GROUP_6:def 3
           .= b * (N * (d * N)) by GROUP_3:10
           .= b * (N * d * N) by GROUP_3:13
           .= b * (d * N * N) by GROUP_3:117
           .= b * (d * N) by GROUP_6:5
           .= b * d * N by GROUP_2:105;
      then
A10:  (d" * b") * (d * b * N) = ((d" * b") * (b * d)) * N by A7,A9,GROUP_2:105
                           .= (d" * (b" * (b * d))) * N by GROUP_1:def 3
                           .= (d" * ((b" * b) * d)) * N by GROUP_1:def 3
                           .= (d" * (1_G * d)) * N by GROUP_1:def 5
                           .= (d" *  d) * N by GROUP_1:def 4
                           .= 1_G * N by GROUP_1:def 5
                           .= carr N by GROUP_2:109;
      (d" * b") * (d * b * N) = (d" * b") * (d * b) * N by GROUP_2:105
                             .= [.d,b.] * N by GROUP_5:16;
      then [.d,b.] in N by A10,GROUP_2:113;
      hence x in carr N by A5,STRUCT_0:def 5;
    end;
    hence thesis;
  end;
  gr carr N = N by GROUP_4:31;
  hence thesis by A4,GROUP_4:32;
end;
