
theorem
  for L being antisymmetric transitive with_suprema RelStr for V being
Subset of L, x, y being Element of L st x <= y holds {x} "\/" V is_finer_than {
  y} "\/" V
proof
  let L be antisymmetric transitive with_suprema RelStr, V be Subset of L, x,
  y be Element of L such that
A1: x <= y;
A2: {x} "\/" V = {x "\/" s where s is Element of L: s in V} by YELLOW_4:15;
  let b be Element of L;
  assume b in {x} "\/" V;
  then consider s being Element of L such that
A3: b = x "\/" s and
A4: s in V by A2;
  take a = y "\/" s;
  {y} "\/" V = {y "\/" t where t is Element of L: t in V} by YELLOW_4:15;
  hence a in {y} "\/" V by A4;
  thus thesis by A1,A3,WAYBEL_1:2;
end;
