
theorem Th10:
  for S being up-complete LATTICE,
  D being directed non empty Subset of S holds lim_inf Net-Str D = sup D
proof
  let S be up-complete LATTICE;
  let D be directed non empty Subset of S;
  set F = (id the carrier of S)|D;
A1: F = id D by FUNCT_3:1;
  lim_inf Net-Str D = sup Net-Str D by Lm6
    .= Sup F by WAYBEL_2:def 1
    .= "\/"(rng F, S) by YELLOW_2:def 5
    .= sup D by A1,RELAT_1:45;
  hence thesis;
end;
