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 Th76:
  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
A1:N1 is Subgroup of N;
   consider N2 be strict normal Subgroup of G such that
A2:the carrier of N2 = N1 ~ N by Th73;
   consider N3 be strict normal Subgroup of G such that
A3:the carrier of N3 = N1 ` N by A1,Th74;
   N3 is Subgroup of N2 by A2,A3,Th55,GROUP_2:57;
  then N3 ~ N c= N2 ~ N by Th57;
  hence thesis by A2,A3;
end;
