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

theorem ::3.15 (2-1), p.72
  for L being complete LATTICE holds L is completely-distributive iff L
is continuous & 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
proof
  let L be complete LATTICE;
  thus L is completely-distributive implies L is continuous & 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
  proof
    assume L is completely-distributive;
    then reconsider L as completely-distributive LATTICE;
    L~ is continuous by Lm3;
    hence thesis by Lm4;
  end;
  thus thesis by Lm2;
end;
