
theorem Th7:
  for L being up-complete Semilattice for D being non empty
  directed Subset of [:L,L:] holds ex_sup_of 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:];
  reconsider D1 = proj1 D, D2 = proj2 D as non empty directed Subset of L by
YELLOW_3:21,22;
  set A = {X where X is non empty directed Subset of L: ex x being Element of
  L st X = {x} "/\" D2 & x in D1};
  union A c= the carrier of L
  proof
    let q be object;
    assume q in union A;
    then consider r being set such that
A1: q in r and
A2: r in A by TARSKI:def 4;
    ex s being non empty directed Subset of L st r = s & ex x being Element
    of L st s = {x} "/\" D2 & x in D1 by A2;
    hence thesis by A1;
  end;
  then reconsider S = union A as Subset of L;
  S = D1 "/\" D2 by Th6;
  hence thesis by WAYBEL_0:75;
end;
