let G be Group; :: thesis: for A being non empty Subset 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 & (N1 ~ A) \/ (N2 ~ A) c= N ~ A )
let A be non empty Subset 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 & (N1 ~ A) \/ (N2 ~ A) c= N ~ A )
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 & (N1 ~ A) \/ (N2 ~ A) c= N ~ A )
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: ( N1 ~ A c= N ~ A & N2 ~ A c= N ~ A ) by Th26;
(N1 ~ A) \/ (N2 ~ A) c= N ~ A
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (N1 ~ A) \/ (N2 ~ A) or x in N ~ A )
assume x in (N1 ~ A) \/ (N2 ~ A) ; :: thesis: x in N ~ A
then ( x in N1 ~ A or x in N2 ~ A ) by XBOOLE_0:def 3;
hence x in N ~ A by A2; :: thesis: verum
end;
hence ex N being strict normal Subgroup of G st
( the carrier of N = N1 * N2 & (N1 ~ A) \/ (N2 ~ A) c= N ~ A ) by A1; :: thesis: verum