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 holds ex N being strict normal
  Subgroup of G st the carrier of N = N1 * N2 & N ` H c= N1 ` H /\ N2 ` 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 is Subgroup of N & N2 is Subgroup of N by A1,Th9; then
A2: N ` H c= N1 ` H & N ` H c= N2 ` H by Th60;
  N ` H c= N1 ` H /\ N2 ` H
  by A2,XBOOLE_0:def 4;
  hence thesis by A1;
end;
