let L be Boolean LATTICE; :: thesis: ( L is arithmetic iff L is algebraic )

thus ( L is arithmetic implies L is algebraic ) ; :: thesis: ( L is algebraic implies L is arithmetic )

assume A1: L is algebraic ; :: thesis: L is arithmetic

then L opp is continuous by Th9, YELLOW_7:38;

then L is completely-distributive by A1, WAYBEL_6:39;

then for x being Element of L ex X being Subset of L st

( X c= ATOM L & x = sup X ) by Lm5;

then ex X being set st L, BoolePoset X are_isomorphic by A1, Lm6;

hence L is arithmetic by Th10, WAYBEL_1:6; :: thesis: verum

thus ( L is arithmetic implies L is algebraic ) ; :: thesis: ( L is algebraic implies L is arithmetic )

assume A1: L is algebraic ; :: thesis: L is arithmetic

then L opp is continuous by Th9, YELLOW_7:38;

then L is completely-distributive by A1, WAYBEL_6:39;

then for x being Element of L ex X being Subset of L st

( X c= ATOM L & x = sup X ) by Lm5;

then ex X being set st L, BoolePoset X are_isomorphic by A1, Lm6;

hence L is arithmetic by Th10, WAYBEL_1:6; :: thesis: verum