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

theorem Th26:
  (the carrier of L) --> {Bottom L} in the carrier of MonSet L
proof
  set s = (the carrier of L) --> {Bottom L};
  ex s9 be Function of L, InclPoset Ids L st s = s9 & s9 is monotone &
  for x be Element of L holds s9.x c= downarrow x
  proof
    reconsider s as Function of L, InclPoset Ids L by Th25;
    take s;
    for x holds s.x c= downarrow x by Lm4;
    hence thesis by Lm5;
  end;
  hence thesis by Def13;
end;
