let G, H be Group; :: thesis: for h being Homomorphism of G,H
for A, B being Subgroup of G st A is Subgroup of B holds
h .: A is Subgroup of h .: B
let h be Homomorphism of G,H; :: thesis: for A, B being Subgroup of G st A is Subgroup of B holds
h .: A is Subgroup of h .: B
let A, B be Subgroup of G; :: thesis: ( A is Subgroup of B implies h .: A is Subgroup of h .: B )
assume A is Subgroup of B ; :: thesis: h .: A is Subgroup of h .: B
then the carrier of A c= the carrier of B by GROUP_2:def 5;
then A1: h .: the carrier of A c= h .: the carrier of B by RELAT_1:123;
the carrier of (h .: B) = h .: the carrier of B by Th8;
then the carrier of (h .: A) c= the carrier of (h .: B) by A1, Th8;
hence h .: A is Subgroup of h .: B by GROUP_2:57; :: thesis: verum