
theorem Th8:
  for L being up-complete Semilattice for D being non empty
directed Subset of [:L,L:] holds ex_sup_of {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
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;
  defpred P[set] means ex x being Element of L st $1 = {x} "/\" D2 & x in D1;
  set A = {sup X where X is non empty directed Subset of L: P[X]};
  A c= the carrier of L
  proof
    let q be object;
    assume q in A;
    then ex X being non empty directed Subset of L st q = sup X & P[X];
    hence thesis;
  end;
  then reconsider A as Subset of L;
  set R = {X where X is non empty directed Subset of L: P[X]};
  union R c= the carrier of L
  proof
    let q be object;
    assume q in union R;
    then consider r being set such that
A1: q in r and
A2: r in R 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 UR = union R as Subset of L;
  set a = sup UR;
A3: ex_sup_of UR,L by Th7;
A4: for b being Element of L st A is_<=_than b holds a <= b
  proof
    let b be Element of L such that
A5: A is_<=_than b;
    UR is_<=_than b
    proof
      let k be Element of L;
      assume k in UR;
      then consider k1 being set such that
A6:   k in k1 and
A7:   k1 in R by TARSKI:def 4;
      consider s being non empty directed Subset of L such that
A8:   k1 = s and
A9:   P[s] by A7;
      consider x being Element of L such that
A10:  s = {x} "/\" D2 and
      x in D1 by A9;
A11:  {x} "/\" D2 = {x "/\" d2 where d2 is Element of L: d2 in D2} by
YELLOW_4:42;
      sup s in A by A9;
      then
A12:  sup s <= b by A5;
      consider d2 being Element of L such that
A13:  k = x "/\" d2 and
      d2 in D2 by A6,A8,A10,A11;
      x "/\" d2 <= sup s by A6,A8,A10,A13,Th1,YELLOW_4:1;
      hence k <= b by A13,A12,YELLOW_0:def 2;
    end;
    hence thesis by A3,YELLOW_0:def 9;
  end;
  A is_<=_than a
  proof
    let b be Element of L;
    assume b in A;
    then consider X being non empty directed Subset of L such that
A14: b = sup X and
A15: P[X];
    ex_sup_of X,L & X in R by A15,WAYBEL_0:75;
    hence b <= a by A3,A14,YELLOW_0:34,ZFMISC_1:74;
  end;
  hence thesis by A4,YELLOW_0:15;
end;
