reserve a for set;

theorem Th4:
  for L being lower-bounded sup-Semilattice
  for AR being auxiliary(iv) Relation of L holds AuxBottom L c= AR
proof
  let L be with_suprema lower-bounded Poset;
  let AR be auxiliary(iv) Relation of L;
  let a,b be object;
  assume
A1: [a,b] in AuxBottom L;
  then reconsider a9 = a, b9 = b as Element of L by ZFMISC_1:87;
  [a9,b9] in AuxBottom L by A1;
  then a9 = Bottom L by Def9;
  then [a9,b9] in AR by Def6;
  hence thesis;
end;
