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

theorem
  for AR being auxiliary Relation of L holds
  [(the carrier of L) --> {Bottom L} , AR-below] in the InternalRel of MonSet L
proof
  let AR be auxiliary Relation of L;
  set c = (the carrier of L) --> {Bottom L}, d = AR-below;
  ex f,g be Function of L, InclPoset Ids L st c = f & d = g &
  c in the carrier of MonSet L & d in the carrier of MonSet L & f <= g
  proof
    reconsider c as Function of L, InclPoset Ids L by Th25;
    take c,d;
    c <= d
    proof
      let x be set;
      assume x in the carrier of L;
      then reconsider x9 = x as Element of L;
      d.x = AR-below x9 by Def12;
      then reconsider dx = d.x9 as Ideal of L;
      reconsider dx9 = dx as Element of InclPoset Ids L;
A1:   c.x c= dx by Lm4;
      take c.x9, dx9;
      thus thesis by A1,YELLOW_1:3;
    end;
    hence thesis by Th18,Th26;
  end;
  hence thesis by Def13;
end;
