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 Th28:
  for M being strict normal Subgroup of G holds M is Subgroup of B
  implies B./.(B,M)`*` is Subgroup of G./.M
proof
  let M be strict normal Subgroup of G;
  set I = B./.(B,M)`*`;
  set J = (B,M)`*`;
  set g = the multF of I;
  set f = the multF of G./.M;
  set X = [: the carrier of I,the carrier of I :];
  assume
A1: M is Subgroup of B;
A2: the carrier of I c= the carrier of G./.M
  proof
    let x be object;
    assume x in the carrier of I;
    then consider a being Element of B such that
A3: x = a * J and
    x = J * a by Th13;
    reconsider b = a as Element of G by GROUP_2:42;
    J = M by A1,Def1;
    then a * J = b * M by Th2;
    hence thesis by A3,Th14;
  end;
A4: now
    let x be object;
    assume
A5: x in dom g;
    then consider y,z being object such that
A6: [y,z] = x by RELAT_1:def 1;
    z in the carrier of I by A5,A6,ZFMISC_1:87;
    then consider b being Element of B such that
A7: z = b * J and
A8: z = J * b by Th13;
    y in the carrier of I by A5,A6,ZFMISC_1:87;
    then consider a being Element of B such that
A9: y = a * J and
    y = J * a by Th13;
    reconsider W1 = y, W2 = z as Element of Cosets J by A9,A7,Th14;
A10: g.x = g.(W1,W2) by A6
      .= (a * J) * (J * b) by A9,A8,Def3
      .= a * J * J * b by GROUP_3:11
      .= a * (J * J) * b by GROUP_4:45
      .= a * J * b by GROUP_2:76
      .= a * (J * b) by GROUP_2:106
      .= a * (b * J) by GROUP_3:117
      .= a * b * J by GROUP_2:105;
    reconsider a9 = a, b9 = b as Element of G by GROUP_2:42;
A11: J = M by A1,Def1;
    then
A12: y = a9 * M by A9,Th2;
    z = b9 * M by A7,A11,Th2;
    then reconsider V1 = y, V2 = z as Element of Cosets M by A12,Th14;
A13: a9 * b9 = a * b by GROUP_2:43;
A14: z = M * b9 by A8,A11,Th2;
    f.x = f.(V1,V2) by A6
      .= (a9 * M) * (M * b9) by A12,A14,Def3
      .= a9 * M * M * b9 by GROUP_3:11
      .= a9 * (M * M) * b9 by GROUP_4:45
      .= a9 * M * b9 by GROUP_2:76
      .= a9 * (M * b9) by GROUP_2:106
      .= a9 * (b9 * M) by GROUP_3:117
      .= a9 * b9 * M by GROUP_2:105;
    hence g.x = f.x by A10,A11,A13,Th2;
  end;
  dom g = X & dom f = [: the carrier of G./.M,the carrier of G./.M :] by
FUNCT_2:def 1;
  then dom g = dom f /\ X by A2,XBOOLE_1:28,ZFMISC_1:96;
  then g = f||the carrier of I by A4,FUNCT_1:46;
  hence thesis by A2,GROUP_2:def 5;
end;
