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;
  consider N2 be strict normal Subgroup of G such that
A1:the carrier of N2 = N ~ N by Th73;
  N is Subgroup of N2 by A1,Th54,GROUP_2:57;
  then N ~ N1 c= N2 ~ N1 by Th57;
  hence thesis by A1;
end;
