let G be Group; :: thesis: for H1, H2 being Subgroup of G st the carrier of H1 c= the carrier of H2 holds
H1 is Subgroup of H2
let H1, H2 be Subgroup of G; :: thesis: ( the carrier of H1 c= the carrier of H2 implies H1 is Subgroup of H2 )
set A = the carrier of H1;
set B = the carrier of H2;
set h = the multF of G;
assume A1: the carrier of H1 c= the carrier of H2 ; :: thesis: H1 is Subgroup of H2
hence the carrier of H1 c= the carrier of H2 ; :: according to GROUP_2:def 5 :: thesis: the multF of H1 = the multF of H2 || the carrier of H1
( the multF of H1 = the multF of G || the carrier of H1 & the multF of H2 = the multF of G || the carrier of H2 & [:the carrier of H1,the carrier of H1:] c= [:the carrier of H2,the carrier of H2:] ) by A1, Def5, ZFMISC_1:119;
hence the multF of H1 = the multF of H2 || the carrier of H1 by FUNCT_1:82; :: thesis: verum