
theorem Th25:
  for S being complete LATTICE, N be net of S holds
  the set of all "/\"({N.i where i is Element of N:i >= j},S)
  where j is Element of N
  is directed non empty Subset of S
proof
  let S be complete LATTICE, N be net of S;
  set X = the set of all "/\"({N.i where i is Element of N:i >= j},S)
  where j is Element of N;
  X c= the carrier of S
  proof
    let x be object;
    assume x in X;
    then ex j being Element of N st
    x = "/\"({N.i where i is Element of N:i >= j},S);
    hence thesis;
  end;
  then reconsider X as Subset of S;
  X is non empty directed by WAYBEL11:30;
  hence thesis;
end;
