
theorem Th25:
  for L being antisymmetric transitive with_infima RelStr for V
being upper Subset of L holds {x where x is Element of L : V "/\" {x} c= V} is
  upper Subset of L
proof
  let L be antisymmetric transitive with_infima RelStr, V be upper Subset of L;
  reconsider G1 = {x where x is Element of L : V "/\" {x} c= V} as Subset of L
  by Lm2;
  G1 is upper
  proof
    let x, y be Element of L;
    assume x in G1;
    then
A1: ex x1 being Element of L st x1 = x & V "/\" {x1} c= V;
    assume x <= y;
    then V "/\" {y} c= V by A1,Th17,Th19;
    hence thesis;
  end;
  hence thesis;
end;
