let L be LATTICE; :: thesis: ( ex X being set ex c being closure Function of (BoolePoset X),(BoolePoset X) st
( c is directed-sups-preserving & L, Image c are_isomorphic ) implies L is algebraic )
given X being set , c being closure Function of (BoolePoset X),(BoolePoset X) such that A1: c is directed-sups-preserving and
A2: L, Image c are_isomorphic ; :: thesis: L is algebraic
( Image c is complete LATTICE & Image c is algebraic LATTICE ) by A1, WAYBEL_8:24, YELLOW_2:35;
hence L is algebraic by A2, Th32, WAYBEL_1:6; :: thesis: verum