let G be Group; for N, N1 being normal Subgroup of G st N1 is Subgroup of N holds
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 )
let N, N1 be normal Subgroup of G; ( N1 is Subgroup of N implies 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 ) )
assume
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 )
then consider N2, N3 being 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
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 )
by A1, A2, A3, XBOOLE_1:1; verum