let G be Group; :: thesis: for H being Subgroup of G
for N1, N2 being strict normal Subgroup of G ex N being strict normal Subgroup of G st
( the carrier of N = N1 * N2 & N ` H c= (N1 ` H) /\ (N2 ` H) )
let H be Subgroup of G; :: thesis: for N1, N2 being strict normal Subgroup of G ex N being strict normal Subgroup of G st
( the carrier of N = N1 * N2 & N ` H c= (N1 ` H) /\ (N2 ` H) )
let N1, N2 be strict normal Subgroup of G; :: thesis: ex N being strict normal Subgroup of G st
( the carrier of N = N1 * N2 & N ` H c= (N1 ` H) /\ (N2 ` H) )
consider N being strict normal Subgroup of G such that
A1: the carrier of N = N1 * N2 by Th8;
( N1 is Subgroup of N & N2 is Subgroup of N ) by A1, Th9;
then A2: ( N ` H c= N1 ` H & N ` H c= N2 ` H ) by Th60;
N ` H c= (N1 ` H) /\ (N2 ` H) by A2, XBOOLE_0:def 4;
hence ex N being strict normal Subgroup of G st
( the carrier of N = N1 * N2 & N ` H c= (N1 ` H) /\ (N2 ` H) ) by A1; :: thesis: verum