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 Th20:
  for G1 being Subgroup of G, N being normal Subgroup of G st
  N is strict Subgroup of G1 & [.G1,(Omega).G.] is strict Subgroup of N holds
  G1./.(G1,N)`*` is Subgroup of center (G./.N)
proof
  let G1 be Subgroup of G;
  let N be normal Subgroup of G;
  assume that
A1: N is strict Subgroup of G1 and
A2: [.G1,(Omega).G.] is strict Subgroup of 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;
  for S1 be Element of G./.N st S1 in J holds S1 in center (G./.N)
  proof
    let S1 be Element of G./.N;
    assume
A4: S1 in J;
    for S be Element of G./.N holds S1 * S = S * S1
    proof
      let S be Element of G./.N;
      consider a being Element of G such that
A5:   S = a * N & S = N * a by GROUP_6:21;
      consider c being Element of G1 such that
A6:   S1 = c * I & S1 = I * c by A3,A4,GROUP_6:23;
      reconsider d = c as Element of G by GROUP_2:42;
A7:   d in G1 by STRUCT_0:def 5;
      a in (Omega).G by STRUCT_0:def 5;
      then [.d,a.] in [.G1,(Omega).G.] by A7,GROUP_5:65; then
A8:   [.d,a.] in N by A2,GROUP_2:40;
A9:   @S = S & @S1 = S1 & c * I = d * N & N * d = I * c by GROUP_6:2; then
A10:  S * S1 = a * N * (d * N) by A5,A6,GROUP_6:def 3
            .= a * d * N by GROUP_11:1;
A11:  S1 * S = d * N * (a * N) by A5,A6,A9,GROUP_6:def 3
            .= d * a * N by GROUP_11:1;
A12:  a * d * [.d,a.] * N = a * d * ([.d,a.] * N) by GROUP_2:32
                         .= a * d *  N by A8,GROUP_2:113;
      a * d * [.d,a.] * N = a * d * ((d" * a") * (d * a)) * N by GROUP_1:def 3
                       .= (a * d * (d" * a")) * (d * a) * N by GROUP_1:def 3
                       .= (a * (d * (d" * a"))) * (d * a) * N by GROUP_1:def 3
                       .= (a * (d * d" * a")) * (d * a) * N by GROUP_1:def 3
                       .= (a * (1_G * a")) * (d * a) * N by GROUP_1:def 5
                       .= (a * a") * (d * a) * N by GROUP_1:def 4
                       .= 1_G * (d * a) * N by GROUP_1:def 5
                       .= d * a * N by GROUP_1:def 4;
      hence thesis by A10,A11,A12;
    end;
    hence thesis by GROUP_5:77;
  end;
  hence thesis by GROUP_2:58;
end;
