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,N2 be normal Subgroup of G st N1 is Subgroup of N2
  ex N3,N4 being strict normal Subgroup of G st
  the carrier of N3 = N ~ N1 & the carrier of N4 = N ~ N2 &
  N3 ~ N1 c= N4 ~ N1
proof
  let N,N1,N2 be normal Subgroup of G;
  assume
A1:N1 is Subgroup of N2;
  consider N3 be strict normal Subgroup of G such that
A2:the carrier of N3 = N ~ N1 by Th73;
  consider N4 be strict normal Subgroup of G such that
A3:the carrier of N4 = N ~ N2 by Th73;
  N3 is Subgroup of N4 by A1,A2,A3,Th56,GROUP_2:57;
  then N3 ~ N1 c= N4 ~ N1 by Th57;
  hence thesis by A2,A3;
end;
