
theorem
  for L be lower-bounded continuous sup-Semilattice for B be with_bottom
  CLbasis of L for x be set holds x in rng baseMap B implies x is Ideal of
  subrelstr B
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
  let x be set;
A1: rng baseMap B is Subset of Ids subrelstr B by YELLOW_1:1;
  assume x in rng baseMap B;
  then x in Ids subrelstr B by A1;
  then x in the set of all  X where X is Ideal of subrelstr B  by
WAYBEL_0:def 23;
  then ex I be Ideal of subrelstr B st x = I;
  hence thesis;
end;
