let W be with_non-empty_element set ; :: thesis: for L being LATTICE holds
( L is Object of (W -SUP(SO)_category) iff ( L is strict & L is complete & the carrier of L in W ) )
let L be LATTICE; :: thesis: ( L is Object of (W -SUP(SO)_category) iff ( L is strict & L is complete & the carrier of L in W ) )
the carrier of (W -SUP(SO)_category) c= the carrier of (W -SUP_category) by ALTCAT_2:def 11;
then ( L in the carrier of (W -SUP(SO)_category) implies L is Object of (W -SUP_category) ) ;
then ( L is Object of (W -SUP(SO)_category) iff L is Object of (W -SUP_category) ) by Def11;
hence ( L is Object of (W -SUP(SO)_category) iff ( L is strict & L is complete & the carrier of L in W ) ) by Th15; :: thesis: verum