
theorem
  for S being non empty Poset, N be monotone reflexive net of S holds
  the set of all "/\"({N.i where i is Element of N: i >= j}, S)
  where j is Element of N
  is directed non empty Subset of S
proof
  let S be non empty Poset, N be monotone reflexive net of S;
  set X = the set of all "/\"({N.i where i is Element of N: i >= j}, S)
  where j is Element of N;
  set jj = the Element of N;
A1: "/\"({N.i where i is Element of N: i >= jj}, S) in X;
  X c= the carrier of S
  proof
    let x be object;
    assume x in X;
    then ex j being Element of N st
    x = "/\"({N.i where i is Element of N: i >= j}, S);
    hence thesis;
  end;
  then reconsider X as non empty Subset of S by A1;
  X is directed
  proof
    let x,y be Element of S;
    assume x in X;
    then consider j1 being Element of N such that
A2: x = "/\"({N.i where i is Element of N: i >= j1}, S);
    assume y in X;
    then consider j2 being Element of N such that
A3: y = "/\"({N.i where i is Element of N: i >= j2}, S);
    reconsider j1,j2 as Element of N;
    [#]N is directed by WAYBEL_0:def 6;
    then consider j being Element of N such that
    j in [#]N and
A4: j >= j1 and
A5: j >= j2;
    set z = "/\"({N.i where i is Element of N: i >= j}, S);
    take z;
    thus z in X;
    deffunc up(Element of N) = {N.i where i is Element of N: i >= $1};
A6: for j being Element of N holds ex_inf_of up(j), S
    proof
      let j be Element of N;
      reconsider j9 = j as Element of N;
      now
        take x = N.j;
        j9 <= j9;
        then
A7:     x in up(j);
        thus x is_<=_than up(j)
        proof
          let y be Element of S;
          assume y in up(j);
          then ex i being Element of N st y = N.i & i >= j;
          hence x <= y by WAYBEL11:def 9;
        end;
        let y be Element of S;
        assume y is_<=_than up(j);
        hence y <= x by A7;
      end;
      hence thesis by YELLOW_0:16;
    end;
    then
A8: ex_inf_of up(j1), S;
A9: ex_inf_of up(j2), S by A6;
A10: ex_inf_of up(j), S by A6;
    set A = {N.i where i is Element of N:i >= j};
    A c= {N.k where k is Element of N:k >= j1}
    proof
      let a be object;
      assume a in A;
      then consider k being Element of N such that
A11:  a = N.k and
A12:  k >= j;
      reconsider k as Element of N;
      k >= j1 by A4,A12,ORDERS_2:3;
      hence thesis by A11;
    end;
    hence z >= x by A2,A8,A10,YELLOW_0:35;
    A c= {N.k where k is Element of N:k >= j2}
    proof
      let a be object;
      assume a in A;
      then consider k being Element of N such that
A13:  a = N.k and
A14:  k >= j;
      reconsider k as Element of N;
      k >= j2 by A5,A14,ORDERS_2:3;
      hence thesis by A13;
    end;
    hence thesis by A3,A9,A10,YELLOW_0:35;
  end;
  hence thesis;
end;
