
theorem Th36:
  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-filtered & c is_a_cluster_point_of N holds c = inf 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-filtered & 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,Lm7,Lm8;
  hence c = inf rng the mapping of N by YELLOW_0:31
    .= inf N by YELLOW_2:def 6;
end;
