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 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 &
  N2 ` N c= N3 ~ 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: N2 ` N c= N3 ` N by Th75;
  N3 ` N c= N3 ~ N by Th55;
  hence thesis by A1,A2,A3,XBOOLE_1:1;
end;
