
theorem Th38:
  for S being compact Hausdorff TopLattice st for x being Element
  of S holds x"/\" is continuous holds for N being net of S st N is
  eventually-directed holds ex_sup_of N & sup N in Lim N
proof
  let S be compact Hausdorff TopLattice such that
A1: for x being Element of S holds x "/\" is continuous;
  let N be net of S such that
A2: N is eventually-directed;
A3: for c, d being Point of S st c is_a_cluster_point_of N & d
  is_a_cluster_point_of N holds c = d
  proof
    let c, d be Point of S such that
A4: c is_a_cluster_point_of N and
A5: d is_a_cluster_point_of N;
    thus c = sup N by A1,A2,A4,Th35
      .= d by A1,A2,A5,Th35;
  end;
  consider c being Point of S such that
A6: c is_a_cluster_point_of N by Th30;
  thus ex_sup_of N
  proof
    reconsider d = c as Element of S;
    set X = rng the mapping of N;
    X is_<=_than d & for b being Element of S st X is_<=_than b holds d <=
    b by A1,A2,A6,Lm5,Lm6;
    hence ex_sup_of rng the mapping of N, S by YELLOW_0:15;
  end;
  c = sup N by A1,A2,A6,Th35;
  hence thesis by A6,A3,Th33;
end;
