
theorem
  for R being order_consistent up-complete /\-complete non empty TopLattice
  for N being eventually-directed net of R, x being Element of R holds
  x <= lim_inf N iff x is_a_cluster_point_of N
proof
  let R be order_consistent up-complete /\-complete non empty TopLattice,
  N be eventually-directed net of R, x be Element of R;
  thus x <= lim_inf N implies x is_a_cluster_point_of N by Th28;
  thus x is_a_cluster_point_of N implies x <= lim_inf N
  proof
    assume
A1: x is_a_cluster_point_of N;
    defpred P[Element of N] means not contradiction;
    deffunc F(Element of N) = "/\"({N.i where i is Element of N:i >= $1}, R);
    set X = {F(j) where j is Element of N: P[j]};
    X is Subset of R from DOMAIN_1:sch 8;
    then reconsider D = X as Subset of R;
    reconsider D as non empty directed Subset of R by Th7;
    for G being Subset of R st G is open holds x in G implies {sup D} meets G
    proof
      let G be Subset of R such that
A2:   G is open;
      assume x in G;
      then reconsider G as a_neighborhood of x by A2,CONNSP_2:3;
A3:   N is_often_in G by A1;
      now
        let i be Element of N;
        consider j1 being Element of N such that
        i <= j1 and
A4:     N.j1 in G by A3;
        consider j2 being Element of N such that
A5:     for k being Element of N st j2 <= k holds N.j1 <= N.k by WAYBEL_0:11;
        defpred P[Element of N] means $1 >= j2;
        deffunc F(Element of N) = N.$1;
        set E = {F(k) where k is Element of N: P[k]};
A6:     E is Subset of R from DOMAIN_1:sch 8;
        consider j9 being Element of N such that
A7:     j9 >= j2 and j9 >= j2 by YELLOW_6:def 3;
        N.j9 in E by A7;
        then reconsider E9 = E as non empty Subset of R by A6;
A8:     ex_inf_of E9,R by WAYBEL_0:76;
        N.j1 is_<=_than E9
        proof
          let b be Element of R;
          assume b in E9;
          then ex k being Element of N st ( b = N.k)&( k >= j2);
          hence N.j1 <= b by A5;
        end;
        then
A9:     N.j1 <= "/\"(E9,R) by A8,YELLOW_0:31;
        for a being Element of R holds downarrow a = Cl {a} by Def2;
        then
A10:    G is upper by A2,Th21;
        then
A11:    "/\"(E9,R) in G by A4,A9;
        "/\"(E9,R) in D;
        then "/\"(E9,R) <= sup D by Th17;
        hence sup D in G by A10,A11;
      end;
      then
A12:  sup D in G;
      sup D in {sup D} by TARSKI:def 1;
      hence thesis by A12,XBOOLE_0:3;
    end;
    then x in Cl {sup D} by PRE_TOPC:24;
    then x in downarrow sup D by Def2;
    hence thesis by WAYBEL_0:17;
  end;
end;
