let L be Boolean LATTICE; :: thesis: ( L is arithmetic iff ( L is continuous & L opp is continuous ) )
thus ( L is arithmetic implies ( L is continuous & L opp is continuous ) ) by Th11, YELLOW_7:38; :: thesis: ( L is continuous & L opp is continuous implies L is arithmetic )
assume ( L is continuous & L opp is continuous ) ; :: thesis: L is arithmetic
then L is completely-distributive by WAYBEL_6:39;
then ( L is complete & ( for x being Element of L ex X being Subset of L st
( X c= ATOM L & x = sup X ) ) ) by Lm6;
then ex X being set st L, BoolePoset X are_isomorphic by Lm7;
hence L is arithmetic by Th12, WAYBEL_1:7; :: thesis: verum