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