
theorem Th6:
  for L being Semilattice, D being non empty directed Subset of [:L
  ,L:] holds 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} = proj1 D "/\" proj2 D
proof
  let L be Semilattice, D be non empty directed Subset of [:L,L:];
  set D1 = proj1 D, D2 = proj2 D;
  defpred P[set] means ex x being Element of L st $1 = {x} "/\" proj2 D & x in
  proj1 D;
  reconsider T = D2 as non empty directed Subset of L by YELLOW_3:21,22;
A1: D1 "/\" D2 = { x "/\" y where x, y is Element of L : x in D1 & y in D2 }
  by YELLOW_4:def 4;
  thus union {X where X is non empty directed Subset of L: P[X]} c= D1 "/\" D2
  proof
    let q be object;
    assume q in union {X where X is non empty directed Subset of L: P[X]};
    then consider w being set such that
A2: q in w and
A3: w in {X where X is non empty directed Subset of L: P[X]} by TARSKI:def 4;
    consider e being non empty directed Subset of L such that
A4: w = e and
A5: P[e] by A3;
    consider p being Element of L such that
A6: e = {p} "/\" D2 and
A7: p in D1 by A5;
    {p} "/\" D2 = { p "/\" y where y is Element of L : y in D2 } by YELLOW_4:42
;
    then ex y being Element of L st q = p "/\" y & y in D2 by A2,A4,A6;
    hence thesis by A1,A7;
  end;
  let q be object;
  assume q in D1 "/\" D2;
  then consider x, y being Element of L such that
A8: q = x "/\" y and
A9: x in D1 and
A10: y in D2 by A1;
  reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5;
  xx "/\" T is non empty directed;
  then
A11: {x} "/\" D2 in {X where X is non empty directed Subset of L: P[X]} by A9;
  {x} "/\" D2 = { x "/\" d where d is Element of L : d in D2 } by YELLOW_4:42;
  then q in {x} "/\" D2 by A8,A10;
  hence thesis by A11,TARSKI:def 4;
end;
