reserve a for set;

theorem
  for L being lower-bounded sup-Semilattice holds
  Top InclPoset Aux L = IntRel L
proof
  let L be lower-bounded sup-Semilattice;
  IntRel L = "/\"({},InclPoset Aux L)
  proof
    set P = InclPoset Aux L;
    set I = IntRel L;
    I in Aux L by Def8;
    then reconsider I as Element of P by YELLOW_1:1;
A1: I is_<=_than {};
    for b being Element of P st b is_<=_than {} holds I >= b
    proof
      let b be Element of P;
      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;
      assume b is_<=_than {};
      b9 c= I by Th2;
      hence thesis by YELLOW_1:3;
    end;
    hence thesis by A1,YELLOW_0:31;
  end;
  hence thesis;
end;
