reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem
  for N1,N2 be strict normal Subgroup of G
  ex N being strict normal Subgroup of G st the carrier of N = N1 * N2 &
  N ~ A c= ((N1 ~ A) * N2) /\ ((N2 ~ A) * N1)
proof
  let N1,N2 be strict normal Subgroup of G;
  consider N be strict normal Subgroup of G such that
A1:the carrier of N = N1 * N2 by Th8;
  N ~ A c= ((N1 ~ A) * N2) /\ ((N2 ~ A) * N1)
  proof
    let x be object;
    assume
A2: x in N ~ A;
    then reconsider x as Element of G;
 x * N meets A by A2,Th14;
    then consider x1 be object such that
A3: x1 in x * N & x1 in A by XBOOLE_0:3;
    reconsider x1 as Element of G by A3;
    consider y be Element of G such that
A4: x1 = x * y & y in N by A3,GROUP_2:103;
A5: y in N1 * N2 by A1,A4,STRUCT_0:def 5;
    then consider a,b be Element of G such that
A6: y = a * b & a in N1 & b in N2 by Th6;
A7: x1 = x * a * b by A4,A6,GROUP_1:def 3;
    a in carr(N1) by A6,STRUCT_0:def 5;
    then
A8: x * a * b in x * N1 * b by GROUP_8:15;
    x * N1 * b = x * (N1 * b) by GROUP_2:106
            .= x * (b * N1) by GROUP_3:117
            .= (x * b) * N1 by GROUP_2:105;
    then (x * b) * N1 meets A by A3,A7,A8,XBOOLE_0:3;
    then
A9: x * b in N1 ~ A;
A10:(x * b) * b" = x * (b * b") by GROUP_1:def 3
            .= x * 1_G by GROUP_1:def 5
            .= x by GROUP_1:def 4;
    b" in N2 by A6,GROUP_2:51;
    then
A11:x in (N1 ~ A) * N2 by A9,A10,GROUP_2:94;
    y in N2 * N1 by A5,GROUP_3:125;
    then consider a,b be Element of G such that
A12: y = a * b & a in N2 & b in N1 by Th6;
A13: x1 = x * a * b by A4,A12,GROUP_1:def 3;
    a in carr(N2) by A12,STRUCT_0:def 5;
    then
A14: x * a * b in x * N2 * b by GROUP_8:15;
    x * N2 * b = x * (N2 * b) by GROUP_2:106
              .= x * (b * N2) by GROUP_3:117
              .= (x * b) * N2 by GROUP_2:105;
    then (x * b) * N2 meets A by A3,A13,A14,XBOOLE_0:3;
    then
A15: x * b in N2 ~ A;
A16:(x * b) * b" = x * (b * b") by GROUP_1:def 3
            .= x * 1_G by GROUP_1:def 5
            .= x by GROUP_1:def 4;
    b" in N1 by A12,GROUP_2:51;
    then x in (N2 ~ A) * N1 by A15,A16,GROUP_2:94;
    hence thesis by A11,XBOOLE_0:def 4;
  end;
  hence thesis by A1;
end;
