
theorem Th10:
  for T being lower complete TopLattice, BB being prebasis of T
  for F being non empty filtered Subset of T st for A being Subset of T st A in
  BB & inf F in A holds F meets A holds inf F in Cl F
proof
  let T be lower complete TopLattice, BB be prebasis of T;
  let F be non empty filtered Subset of T such that
A1: for A being Subset of T st A in BB & inf F in A holds F meets A;
  set N = F opp+id, x = inf F;
A2: for A being Subset of T st A in BB & x in A holds N is_eventually_in A
  proof
    let A be Subset of T;
    assume that
A3: A in BB and
A4: x in A;
    A is open by A3,TOPS_2:def 1;
    then
A5: A is lower by Th5;
    F meets A by A3,A4,A1;
    then consider i being object such that
A6: i in F and
A7: i in A by XBOOLE_0:3;
    reconsider i as Element of N by A6,YELLOW_9:7;
    take i;
    let j be Element of N;
A8: N is full SubRelStr of T opp by YELLOW_9:7;
    then reconsider a = i, b = j as Element of T opp by YELLOW_0:58;
    assume i <= j;
    then a <= b by A8,YELLOW_0:59;
    then
A9: ~b <= ~a by YELLOW_7:1;
    N.j = j by YELLOW_9:7;
    hence thesis by A9,A7,A5;
  end;
A10: the carrier of N = F by YELLOW_9:7;
  rng netmap(N,T) c= F
  proof
    let x be object;
    assume x in rng netmap(N,T);
    then consider a being object such that
A11: a in dom netmap(N,T) and
A12: x = (netmap(N,T)).a by FUNCT_1:def 3;
    reconsider a as Element of N by A11;
    x = N.a by A12
      .= a by YELLOW_9:7;
    hence thesis by A10;
  end;
  hence thesis by A2,YELLOW_9:39;
end;
