
theorem
  for L being complete Semilattice holds L is meet-continuous iff for x
  being Element of L, J being set for f being Function of J,the carrier of L
  holds x "/\" Sup f = sup(x "/\" FinSups f)
proof
  let L be complete Semilattice;
  hereby
    assume L is meet-continuous;
    then 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))
by Th54;
    hence for x being Element of L, J being set for f being Function of J,the
    carrier of L holds x "/\" Sup f = sup(x "/\" FinSups f) by Th47;
  end;
  assume for x being Element of L, J being set for f being Function of J,the
  carrier of L holds x "/\" Sup f = sup(x "/\" FinSups f);
  then for x being Element of L, N being prenet of L st N is
  eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) by
Th48;
  hence L is up-complete & L is satisfying_MC by Th42;
end;
