
theorem Th37:
  for S being Hausdorff TopLattice st (for N being net of S st N
  is eventually-directed holds ex_sup_of N & sup N in Lim N) & (for x being
  Element of S holds x"/\" is continuous) holds S is meet-continuous
proof
  let S be Hausdorff TopLattice such that
A1: for N being net of S st N is eventually-directed holds ex_sup_of N &
  sup N in Lim N and
A2: for x being Element of S holds x "/\" is continuous;
  for X being non empty directed Subset of S holds ex_sup_of X,S
  proof
    let X be non empty directed Subset of S;
    reconsider n = id X as Function of X, the carrier of S by FUNCT_2:7;
    set N = NetStr (#X,(the InternalRel of S)|_2 X,n#);
    N is eventually-directed by WAYBEL_2:20;
    then
A3: ex_sup_of N by A1;
    rng the mapping of N = X by RELAT_1:45;
    hence thesis by A3;
  end;
  hence S is up-complete by WAYBEL_0:75;
  for x being Element of S, M being net of S st M is eventually-directed
  holds x "/\" sup M = sup ({x} "/\" rng netmap (M,S))
  proof
    let x be Element of S, M be net of S such that
A4: M is eventually-directed;
A5: sup M in Lim M by A1,A4;
    then reconsider M1 = M as convergent net of S by YELLOW_6:def 16;
    set xM = x "/\" M;
    x "/\" M is eventually-directed by A4,WAYBEL_2:27;
    then
A6: sup xM in Lim xM by A1;
    x"/\" is continuous by A2;
    then
A7: x "/\" lim M1 in Lim (x "/\" M1) by Th26;
    reconsider xM as convergent net of S by A6,YELLOW_6:def 16;
    thus x "/\" sup M = x "/\" lim M1 by A5,YELLOW_6:def 17
      .= lim xM by A7,YELLOW_6:def 17
      .= Sup the mapping of xM by A6,YELLOW_6:def 17
      .= sup rng the mapping of xM by YELLOW_2:def 5
      .= sup ({x} "/\" rng netmap (M,S)) by WAYBEL_2:23;
  end;
  hence thesis by Th9;
end;
