reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem
  for M being strict normal Subgroup of G
  for J being strict normal Subgroup of G./.M st
    J = N./.(N,M)`*` & M is Subgroup of N holds
      (G./.M)./.J,G./.N are_isomorphic
proof
  let M be strict normal Subgroup of G;
  let J be strict normal Subgroup of G./.M;
  assume that
A1: J = N./.(N,M)`*` and
A2: M is Subgroup of N;
  defpred P[set,set] means for a st $1 = a * M holds $2 = a * N;
A3: for x being Element of G./.M ex y being Element of G./.N st P[x,y]
  proof
    let x be Element of G./.M;
    consider a such that
A4: x = a * M and
    x = M * a by Th13;
    reconsider y = a * N as Element of G./.N by Th14;
    take y;
    let b;
    assume x = b * M;
    then a" * b in M by A4,GROUP_2:114;
    then a" * b in N by A2,GROUP_2:40;
    hence thesis by GROUP_2:114;
  end;
  consider f being Function of G./.M, G./.N such that
A5: for x being Element of G./.M holds P[x,f.x] from FUNCT_2:sch 3(A3);
  now
    let x,y be Element of G./.M;
    consider a such that
A6: x = a * M and
    x = M * a by Th13;
    consider b such that
A7: y = b * M and
    y = M * b by Th13;
A8: x * y = @x * @y by Def3
      .= a * M * b * M by A6,A7,GROUP_3:9
      .= a * (M * b) * M by GROUP_2:106
      .= a * (b * M) * M by GROUP_3:117
      .= a * ((b * M) * M) by GROUP_2:96
      .= a * (b * M) by Th5
      .= a * b * M by GROUP_2:105;
A9: f.y = b * N by A5,A7;
A10: f.x = a * N by A5,A6;
    f.x * f.y = @(f.x) * @(f.y) by Def3
      .= a * N * b * N by A10,A9,GROUP_3:9
      .= a * (N * b) * N by GROUP_2:106
      .= a * (b * N) * N by GROUP_3:117
      .= a * ((b * N) * N) by GROUP_2:96
      .= a * (b * N) by Th5
      .= a * b * N by GROUP_2:105;
    hence f.(x * y) = f.x * f.y by A5,A8;
  end;
  then reconsider f as Homomorphism of G./.M,G./.N by Def6;
A11: Ker f = J
  proof
    let S be Element of G./.M;
    thus S in Ker f implies S in J
    proof
      assume S in Ker f;
      then
A12:  f.S = 1_(G./.N) by Th41
        .= carr N by Th24;
      consider a such that
A13:  S = a * M and
A14:  S = M * a by Th13;
      f.S = a * N by A5,A13;
      then a in N by A12,GROUP_2:113;
      then reconsider q = a as Element of N;
      (N,M)`*` = M by A2,Def1;
      then S = q * (N,M)`*` & S = (N,M)`*` * q by A13,A14,Th2;
      hence thesis by A1,Th23;
    end;
    assume S in J;
    then consider a being Element of N such that
A15: S = a * (N,M)`*` and
    S = (N,M)`*` * a by A1,Th23;
    reconsider a9 = a as Element of G by GROUP_2:42;
    (N,M)`*` = M by A2,Def1;
    then S = a9 * M by A15,Th2;
    then
A16: f.S = a9 * N by A5;
    a in N;
    then f.S = carr N by A16,GROUP_2:113
      .= 1_(G./.N) by Th24;
    hence thesis by Th41;
  end;
  the carrier of G./.N c= rng f
  proof
    let x be object;
    assume x in the carrier of G./.N;
    then x in G./.N;
    then consider a such that
A17: x = a * N and
    x = N * a by Th23;
    reconsider S = a * M as Element of G./.M by Th14;
    f.S = a * N & dom f = the carrier of G./.M by A5,FUNCT_2:def 1;
    hence thesis by A17,FUNCT_1:def 3;
  end;
  then rng f = the carrier of G./.N;
  then f is onto;
  then Image f = G./.N by Th57;
  hence thesis by A11,Lm3;
end;
