
theorem Th9:
  for L being non empty reflexive transitive RelStr st for x being
Element of L, N being net of L st N is eventually-directed holds x "/\" sup N =
  sup ({x} "/\" rng netmap (N,L)) holds L is satisfying_MC
proof
  let L be non empty reflexive transitive RelStr such that
A1: for x being Element of L, N being net of L st N is
  eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L));
  let x be Element of L, D be non empty directed Subset of L;
  reconsider n = id D as Function of D, the carrier of L by FUNCT_2:7;
  set N = NetStr (#D,(the InternalRel of L)|_2 D,n#);
A2: N is eventually-directed & Sup netmap (N,L) = sup N by WAYBEL_2:20;
  D c= D;
  then
A3: D = n.:D by FUNCT_1:92
    .= rng netmap (N,L) by RELSET_1:22;
  hence x "/\" sup D = x "/\" Sup netmap (N,L) by YELLOW_2:def 5
    .= sup ({x} "/\" D) by A1,A3,A2;
end;
