
theorem Th9:
  for L being up-complete Semilattice for D being non empty
  directed Subset of [:L,L:] holds "\/" ({ 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 },L
  ) <= "\/" (union {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},L)
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;
A1: "\/"(union {X where X is non empty directed Subset of L: P[X]},L)
  is_>=_than { sup X where X is non empty directed Subset of L: P[X] }
  proof
    let a be Element of L;
    assume a in { sup X where X is non empty directed Subset of L: P[X] };
    then consider q being non empty directed Subset of L such that
A2: a = sup q and
A3: P[q];
A4: q in {X where X is non empty directed Subset of L: P[X]} by A3;
    ex_sup_of q,L & ex_sup_of union {X where X is non empty directed
    Subset of L : P[X]},L by Th7,WAYBEL_0:75;
    hence
    a <= "\/"(union {X where X is non empty directed Subset of L: P[X]},L
    ) by A2,A4,YELLOW_0:34,ZFMISC_1:74;
  end;
  ex_sup_of { sup X where X is non empty directed Subset of L: P[X] },L by Th8;
  hence thesis by A1,YELLOW_0:def 9;
end;
