
theorem Th17:
  for L being antisymmetric transitive with_infima RelStr for V
  being Subset of L, x, y being Element of L st x <= y holds {y} "/\" V
  is_coarser_than {x} "/\" V
proof
  let L be antisymmetric transitive with_infima RelStr, V be Subset of L, x, y
  be Element of L such that
A1: x <= y;
A2: {y} "/\" V = {y "/\" s where s is Element of L: s in V} by YELLOW_4:42;
  let b be Element of L;
  assume b in {y} "/\" V;
  then consider s being Element of L such that
A3: b = y "/\" s and
A4: s in V by A2;
  take a = x "/\" s;
  {x} "/\" V = {x "/\" t where t is Element of L: t in V} by YELLOW_4:42;
  hence a in {x} "/\" V by A4;
  thus thesis by A1,A3,WAYBEL_1:1;
end;
