reserve a for set;

theorem
  for L being lower-bounded sup-Semilattice holds
  Bottom InclPoset Aux L = AuxBottom L
proof
  let L be with_suprema lower-bounded Poset;
  AuxBottom L in Aux L by Def8;
  then reconsider N = AuxBottom L as Element of InclPoset Aux L by YELLOW_1:1;
A1: N is_>=_than {};
  for b being Element of InclPoset Aux L st b is_>=_than {} holds N <= b
  proof
    let b be Element of InclPoset Aux L;
    assume b is_>=_than {};
    b in the carrier of InclPoset Aux L;
    then b in Aux L by YELLOW_1:1;
    then reconsider b9 = b as auxiliary Relation of L by Def8;
    N c= b9 by Th4;
    hence thesis by YELLOW_1:3;
  end;
  hence thesis by A1,YELLOW_0:30;
end;
