let G be Group; :: thesis: for H, N being normal Subgroup of G ex M being strict normal Subgroup of G st the carrier of M = N ~ H
let H, N be normal Subgroup of G; :: thesis: ex M being strict normal Subgroup of G st the carrier of M = N ~ H
consider M being strict Subgroup of G such that
A1: the carrier of M = N ~ H by Th71;
for x being Element of G holds x * M c= M * x
proof
let x be Element of G; :: thesis: x * M c= M * x
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in x * M or y in M * x )
assume A2: y in x * M ; :: thesis: y in M * x
then reconsider y = y as Element of G ;
consider z being Element of G such that
A3: ( y = x * z & z in M ) by ;
z in the carrier of M by ;
then consider z9 being Element of G such that
A4: ( z9 in z * N & z9 in H ) by ;
(x * z9) * (x ") in H by ;
then A5: (x * z9) * (x ") in carr H by STRUCT_0:def 5;
A6: (x * z9) * (x ") in (x * (z * N)) * (x ") by ;
(x * (z * N)) * (x ") = x * ((z * N) * (x ")) by GROUP_2:33
.= x * (z * (N * (x "))) by GROUP_2:33
.= x * (z * ((x ") * N)) by GROUP_3:117
.= x * ((z * (x ")) * N) by GROUP_2:32
.= (x * (z * (x "))) * N by GROUP_2:32
.= ((x * z) * (x ")) * N by GROUP_1:def 3 ;
then ((x * z) * (x ")) * N meets carr H by ;
then (x * z) * (x ") in N ~ H ;
then A7: (x * z) * (x ") in M by ;
((x * z) * (x ")) * x = y by ;
hence y in M * x by ; :: thesis: verum
end;
then M is normal Subgroup of G by GROUP_3:118;
hence ex M being strict normal Subgroup of G st the carrier of M = N ~ H by A1; :: thesis: verum