
theorem :: THEOREM 4.18. (2) iff (6)
  for L be Boolean LATTICE holds L is arithmetic iff L is
  completely-distributive
proof
  let L be Boolean LATTICE;
  thus L is arithmetic implies L is completely-distributive
  proof
    assume
A1: L is arithmetic;
    then L opp is continuous by Th9,YELLOW_7:38;
    hence thesis by A1,WAYBEL_6:39;
  end;
  assume
A2: L is completely-distributive;
  then
  for x be Element of L ex X be Subset of L st X c= ATOM L & x = sup X by Lm5;
  then ex X be set st L,BoolePoset X are_isomorphic by A2,Lm6;
  hence thesis by Th10,WAYBEL_1:6;
end;
