
theorem Th35:
  for S being compact Hausdorff TopLattice, c being Point of S for
N being net of S st (for x being Element of S holds x"/\" is continuous) & N is
  eventually-directed & c is_a_cluster_point_of N holds c = sup N
proof
  let S be compact Hausdorff TopLattice, c be Point of S, N be net of S such
  that
A1: ( for x being Element of S holds x"/\" is continuous)& N is
  eventually-directed & c is_a_cluster_point_of N;
  reconsider c9 = c as Element of S;
  c9 is_>=_than rng the mapping of N & for b being Element of S st rng the
  mapping of N is_<=_than b holds c9 <= b by A1,Lm5,Lm6;
  hence c = sup rng the mapping of N by YELLOW_0:30
    .= sup N by YELLOW_2:def 5;
end;
