
theorem Th4:
  for L being non empty transitive RelStr, A, B being Subset of L
  st A is_coarser_than B holds uparrow A c= uparrow 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 uparrow 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 16;
  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 16;
end;
