
theorem
  for L being non empty RelStr, V, S, T being Subset of L st S
  is_finer_than T & V is lower & T c= V holds S c= V
proof
  let L be non empty RelStr, V, S, T be Subset of L such that
A1: S is_finer_than T and
A2: V is lower & T c= V;
  let q be object;
  assume
A3: q in S;
  then reconsider S1 = S as non empty Subset of L;
  reconsider a = q as Element of S1 by A3;
  ex b being Element of L st b in T & a <= b by A1;
  hence thesis by A2;
end;
