
theorem Th17:
  for L1 be continuous lower-bounded sup-Semilattice st L1 is
  infinite for B1 be with_bottom CLbasis of L1 holds B1 is infinite
proof
  let L1 be continuous lower-bounded sup-Semilattice;
  assume
A1: L1 is infinite;
  let B1 be with_bottom CLbasis of L1;
  dom supMap subrelstr B1 = Ids subrelstr B1 & rng supMap subrelstr B1 =
  the carrier of L1 by WAYBEL23:51,65;
  then card the carrier of L1 c= card Ids subrelstr B1 by CARD_1:12;
  then Ids subrelstr B1 is infinite by A1;
  then subrelstr B1 is infinite by Th16;
  hence thesis by YELLOW_0:def 15;
end;
