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;
reserve N1,N2 for Subgroup 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 & N1 ~ H * N2 ~ H c= N ~ H
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;
  N1 ~ H * N2 ~ H c= N ~ H
  proof
    let x be object;
    assume
A2: x in N1 ~ H * N2 ~ H;
    then reconsider x as Element of G;
    consider a,b be Element of G such that
A3: x = a * b & a in N1 ~ H & b in N2 ~ H by A2;
A4: a * N1 meets carr(H) by A3,Th51;
A5: b * N2 meets carr(H) by A3,Th51;
    consider x1 be object such that
A6: x1 in a * N1 & x1 in carr(H) by A4,XBOOLE_0:3;
    consider x2 be object such that
A7: x2 in b * N2 & x2 in carr(H) by A5,XBOOLE_0:3;
    reconsider x1 as Element of G by A6;
    reconsider x2 as Element of G by A7;
A8: x1 * x2 in carr(H) * carr(H)  by A6,A7;
A9: x1 * x2 in (a * N1) * (b * N2) by A6,A7;
    carr(H) * carr(H) = carr(H) by GROUP_2:76; then
A10:(a * N1) * (b * N2) meets carr(H) by A8,A9,XBOOLE_0:3;
    (a * N1) * (b * N2) = (a * b) * N by A1,Th10;
    hence thesis by A3,A10;
  end;
  hence thesis by A1;
end;
