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 ` A
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 ` A c= N1 ` A & N ` A c= N2 ` A by Th25;
  N ` A c= N1 ` A /\ N2 ` A
  by A2,XBOOLE_0:def 4;
  hence thesis by A1;
end;
