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 Th25:
  for G,H being strict Group,h being Homomorphism of G,H
  for G1 being strict Subgroup of G
  for G2 being strict normal Subgroup of G
  for H1 being strict Subgroup of Image h
  for H2 being strict normal Subgroup of Image h
   st G2 is strict Subgroup of G1 &
   G1./.(G1,G2)`*` is Subgroup of center (G./.G2)
   & H1=h.:G1 & H2=h.:G2 holds
  H1./.(H1,H2)`*` is Subgroup of center (Image h./.H2)
proof
  let G,H be strict Group;
  let h be Homomorphism of G,H;
  let G1 be strict Subgroup of G;
  let G2 be strict normal Subgroup of G;
  let H1 be strict Subgroup of Image h;
  let H2 be strict normal Subgroup of Image h;
  assume that
A1: G2 is strict Subgroup of G1 and
A2: G1./.(G1,G2)`*` is Subgroup of center (G./.G2) and
A3: H1=h.:G1 & H2=h.:G2;
A4:  H2 is strict Subgroup of H1 by A1,A3,GRSOLV_1:12; then
A5: (H1,H2)`*` = H2 by GROUP_6:def 1;
  then  reconsider I = H2 as normal Subgroup of H1;
  reconsider J = H1./.(H1,H2)`*` as Subgroup of Image h./.H2 by A4,GROUP_6:28;
  for T be Element of Image h./.H2 st T in J holds T in center (Image h./.H2)
  proof
    let T be Element of Image h./.H2;
    assume
A6: T in J;
    for S be Element of Image h./.H2 holds S * T = T * S
    proof
      let S be Element of Image h./.H2;
      consider g being Element of Image h such that
A7:   S = g * H2 & S = H2 * g by GROUP_6:21;
      consider h1 being Element of H1 such that
A8:   T = h1 * I & T = I * h1 by A5,A6,GROUP_6:23;
      reconsider h2 = h1 as Element of Image h by GROUP_2:42;
A9:   @S = S & @T = T & h1 * I = h2 * H2 by GROUP_6:2;
      then
A10:   S * T = g * H2 * (h2 * H2) by A7,A8,GROUP_6:def 3
           .= g * h2 * H2 by GROUP_11:1;
A11:  T * S = h2 * H2 * (g * H2) by A7,A8,A9,GROUP_6:def 3
           .= h2 * g * H2 by GROUP_11:1;
      g in Image h by STRUCT_0:def 5;
      then consider a being Element of G such that
A12:  g = h.a by GROUP_6:45;
A13:  a in (Omega).G by STRUCT_0:def 5;
      h1 in H1 by STRUCT_0:def 5;
      then consider a1 being Element of G1 such that
A14:  h1 = (h|G1).a1 by A3,GROUP_6:45;
A15:  a1 in G1 by STRUCT_0:def 5;
      reconsider a2 = a1 as Element of G by GROUP_2:42;
A16:  h2 = h.a2 by A14,FUNCT_1:49;
then A17:  g * h2 * H2 = h.a * h.a2 * h.:G2 by A12,A3,Th24
                 .= h.:(a * a2 * G2) by Th23;
A18:  h2 * g * H2 = h.a2 * h.a * h.:G2 by A12,A16,A3,Th24
                 .= h.:(a2 * a * G2) by Th23;
A19:  [.G1, (Omega).G.] is strict Subgroup of G2 by A1,A2,Th19;
      [.a2,a.] in [.G1, (Omega).G.] by A13,A15,GROUP_5:65; then
  [.a2,a.] in G2 by A19,GROUP_2:40;
      then a * a2 * G2= a * a2 * ([.a2,a.] * G2) by GROUP_2:113
                .= a * a2 * ((a2" * a") * (a2 * a) * G2) by GROUP_5:16
                .= a * a2 * ((a2" * a") * (a2 * a)) * G2 by GROUP_2:32
                .= (a * a2 * (a2" * a")) * (a2 * a) * G2 by GROUP_1:def 3
                .= (a * (a2 * (a2" * a"))) * (a2 * a) * G2 by GROUP_1:def 3
                .= (a * (a2 * a2" * a")) * (a2 * a) * G2 by GROUP_1:def 3
                .= (a * (1_G * a")) * (a2 * a) * G2 by GROUP_1:def 5
                .= (a * a") * (a2 * a) * G2 by GROUP_1:def 4
                .= 1_G * (a2 * a) * G2 by GROUP_1:def 5
                .= a2 * a * G2 by GROUP_1:def 4;
      hence thesis by A10,A11,A17,A18;
    end;
    hence thesis by GROUP_5:77;
  end;
  hence thesis by GROUP_2:58;
end;
