let L be Boolean LATTICE; :: thesis: ( L is arithmetic iff L is completely-distributive )
thus ( L is arithmetic implies L is completely-distributive ) :: thesis: ( L is completely-distributive implies L is arithmetic )

proof

assume A1: L is arithmetic ; :: thesis: L is completely-distributive
then L opp is continuous by Th9, YELLOW_7:38;
hence L is completely-distributive by A1, WAYBEL_6:39; :: thesis: verum

end;

assume A2: L is completely-distributive ; :: thesis: L is arithmetic
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 A2, Lm6;
hence L is arithmetic by Th10, WAYBEL_1:6; :: thesis: verum