theorem
  for N,N1 be normal Subgroup of G st N1 is Subgroup of N
  ex N2,N3 being strict normal Subgroup of G st
  the carrier of N2 = N1 ~ N & the carrier of N3 = N1 ` N &
  N3 ` N c= N2 ~ N
proof
  let N,N1 be normal Subgroup of G;
  assume N1 is Subgroup of N;
  then consider N2,N3 be strict normal Subgroup of G such that
A1:the carrier of N2 = N1 ~ N and
A2:the carrier of N3 = N1 ` N and
A3: N3 ~ N c= N2 ~ N by Th76;
   N3 ` N c= N3 ~ N by Th55;
  hence thesis by A1,A2,A3,XBOOLE_1:1;
end;
