
theorem
  for L being non empty transitive RelStr, A, B being Subset of L st A
  is_finer_than B holds downarrow A c= downarrow B
proof
  let L be non empty transitive RelStr, A, B be Subset of L such that
A1: for a being Element of L st a in A ex b being Element of L st b in B
  & b >= a;
  let q be object;
  assume
A2: q in downarrow A;
  then reconsider q1 = q as Element of L;
  consider a being Element of L such that
A3: a >= q1 and
A4: a in A by A2,WAYBEL_0:def 15;
  consider b being Element of L such that
A5: b in B and
A6: b >= a by A1,A4;
  b >= q1 by A3,A6,ORDERS_2:3;
  hence thesis by A5,WAYBEL_0:def 15;
end;
