
theorem
  for L being RelStr, A, B being Subset of L st A is_coarser_than B & B
  is upper holds A c= B
proof
  let L be RelStr, A, B be Subset of L such that
A1: A is_coarser_than B and
A2: B is upper;
  let a be object;
  assume
A3: a in A;
  then reconsider a1 = a as Element of L;
  ex b being Element of L st b in B & b <= a1 by A1,A3;
  hence thesis by A2;
end;
