reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th25:
  (the carrier of L) --> {Bottom L} is Function of L, InclPoset Ids L
proof
  {Bottom L} = downarrow Bottom L by Th23;
  then {Bottom L} in the set of all X where X is Ideal of L;
  then {Bottom L} in the carrier of InclPoset Ids L by YELLOW_1:1;
  hence thesis by FUNCOP_1:45;
end;
