reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem ::3.15 (2-3), p.72
  for L being complete LATTICE holds L is completely-distributive iff L
  is distributive continuous & L opp is continuous
proof
  let L be complete LATTICE;
  thus L is completely-distributive implies L is distributive continuous & L~
  is continuous by Lm3;
  assume that
A1: L is distributive continuous and
A2: L~ is continuous;
  reconsider L as distributive continuous LATTICE by A1;
  for l being Element of L ex X being Subset of L st l = sup X & for x
  being Element of L st x in X holds x is co-prime by A2,Lm4;
  hence thesis by Lm2;
end;
