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;

theorem
  for N,M being strict normal Subgroup of G holds M is Subgroup of N
  implies N./.(N,M)`*` is normal Subgroup of G./.M
proof
  let N,M be strict normal Subgroup of G;
  assume
A1: M is Subgroup of N;
  then
A2: (N,M)`*` = M by Def1;
  then reconsider I = M as normal Subgroup of N;
  reconsider J = N./.(N,M)`*` as Subgroup of G./.M by A1,Th28;
  now
    let S be Element of G./.M;
    thus S * J c= J * S
    proof
      let x be object;
      assume x in S * J;
      then consider T being Element of G./.M such that
A3:   x = S * T and
A4:   T in J by GROUP_2:103;
      consider c being Element of N such that
      T = c * I and
A5:   T = I * c by A2,A4,Th23;
      reconsider d = c as Element of G by GROUP_2:42;
      consider a such that
      S = a * M and
A6:   S = M * a by Th13;
      set e = a * (d * a");
      c in N & e = d |^ a" by Th4;
      then e in N by GROUP_5:3;
      then reconsider f = e as Element of N;
A7:   M * e = I * f by Th2;
      reconsider V = I * f as Element of J by A2,Th14;
A8:   V in J;
      reconsider V as Element of G./.M by GROUP_2:42;
      M * d = I * c by Th2;
      then x = M * a * (M * d) by A3,A6,A5,Def3
        .= M * a * (M * d * 1_G) by GROUP_2:37
        .= M * a * (M * d * (a" * a)) by GROUP_1:def 5
        .= M * a * (M * (d * (a" * a))) by GROUP_2:107
        .= M * a * M * (d * (a" * a)) by GROUP_3:11
        .= M * (a * M) * (d * (a" * a)) by GROUP_3:13
        .= M * (M * a) * (d * (a" * a)) by GROUP_3:117
        .= M * ((M * a) * (d * (a" * a))) by GROUP_2:98
        .= M * (M * (a * (d * (a" * a)))) by GROUP_2:107
        .= M * (M * (a * (d * a" * a))) by GROUP_1:def 3
        .= M * (M * (a * (d * a") * a)) by GROUP_1:def 3
        .= M * (M * e * a) by GROUP_2:107
        .= M * (e * M * a) by GROUP_3:117
        .= M * (e * (M * a)) by GROUP_2:106
        .= M * e * (M * a) by GROUP_3:12
        .= V * S by A6,A7,Def3;
      hence thesis by A8,GROUP_2:104;
    end;
  end;
  hence thesis by GROUP_3:118;
end;
