
theorem Th41:
  for L being non empty reflexive RelStr st L is satisfying_MC
  holds for x being Element of L, N being non empty prenet of L st N is
  eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L))
proof
  let L be non empty reflexive RelStr such that
A1: L is satisfying_MC;
  let x be Element of L, N be non empty prenet of L;
  assume N is eventually-directed;
  then
A2: rng netmap (N,L) is directed by Th18;
  thus x "/\" sup N = x "/\" sup rng netmap (N,L) by YELLOW_2:def 5
    .= sup ({x} "/\" rng netmap (N,L)) by A1,A2;
end;
