let G be Group; :: thesis: for N, N1 being normal Subgroup of G ex N2 being strict normal Subgroup of G st

( the carrier of N2 = N ~ N & N ~ N1 c= N2 ~ N1 )

let N, N1 be normal Subgroup of G; :: thesis: ex N2 being strict normal Subgroup of G st

( the carrier of N2 = N ~ N & N ~ N1 c= N2 ~ N1 )

consider N2 being strict normal Subgroup of G such that

A1: the carrier of N2 = N ~ N by Th73;

N is Subgroup of N2 by A1, Th54, GROUP_2:57;

then N ~ N1 c= N2 ~ N1 by Th57;

hence ex N2 being strict normal Subgroup of G st

( the carrier of N2 = N ~ N & N ~ N1 c= N2 ~ N1 ) by A1; :: thesis: verum

( the carrier of N2 = N ~ N & N ~ N1 c= N2 ~ N1 )

let N, N1 be normal Subgroup of G; :: thesis: ex N2 being strict normal Subgroup of G st

( the carrier of N2 = N ~ N & N ~ N1 c= N2 ~ N1 )

consider N2 being strict normal Subgroup of G such that

A1: the carrier of N2 = N ~ N by Th73;

N is Subgroup of N2 by A1, Th54, GROUP_2:57;

then N ~ N1 c= N2 ~ N1 by Th57;

hence ex N2 being strict normal Subgroup of G st

( the carrier of N2 = N ~ N & N ~ N1 c= N2 ~ N1 ) by A1; :: thesis: verum