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 N,N1 be normal Subgroup of G ex N2 being strict normal Subgroup of G st
  the carrier of N2 = N ` N & N ` N1 c= N2 ` N1
proof
  let N,N1 be normal Subgroup of G;
  N is Subgroup of N by GROUP_2:54;
  then consider N2 be strict normal Subgroup of G such that
A1:the carrier of N2 = N ` N by Th74;
  N2 is Subgroup of N by A1,Th53,GROUP_2:57;
  then N ` N1 c= N2 ` N1 by Th60;
  hence thesis by A1;
end;
