
theorem Th5:
  for L being up-complete Semilattice for D being non empty
directed Subset of [:L,L:] holds { sup ({x} "/\" proj2 D) where x is Element of
L: x in proj1 D } = { sup X where X is non empty directed Subset of L: ex x
  being Element of L st X = {x} "/\" proj2 D & x in proj1 D }
proof
  let L be up-complete Semilattice, D be non empty directed Subset of [:L,L:];
  defpred P[set] means ex x being Element of L st $1 = {x} "/\" proj2 D & x in
  proj1 D;
  reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by
YELLOW_3:21,22;
  thus { sup ({x} "/\" proj2 D) where x is Element of L : x in proj1 D } c= {
  sup X where X is non empty directed Subset of L: P[X] }
  proof
    let q be object;
    assume
    q in {sup ({x} "/\" proj2 D) where x is Element of L : x in proj1 D};
    then consider x being Element of L such that
A1: q = sup ({x} "/\" D2) & x in D1;
    reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5;
    xx "/\" D2 is non empty directed;
    hence thesis by A1;
  end;
  let q be object;
  assume q in { sup X where X is non empty directed Subset of L: P[X] };
  then ex X being non empty directed Subset of L st q = sup X & P[X];
  hence thesis;
end;
