
theorem Th24:
  for L being antisymmetric transitive with_infima RelStr, V being
  Subset of L holds {x where x is Element of L : V "/\" {x} c= V} is filtered
  Subset of L
proof
  let L be antisymmetric transitive with_infima RelStr, V be Subset of L;
  reconsider G1 = {x where x is Element of L : V "/\" {x} c= V} as Subset of L
  by Lm2;
  G1 is filtered
  proof
    let x, y be Element of L;
    assume x in G1;
    then consider x1 being Element of L such that
A1: x = x1 and
A2: V "/\" {x1} c= V;
    assume y in G1;
    then consider y1 being Element of L such that
A3: y = y1 and
A4: V "/\" {y1} c= V;
    take z = x1 "/\" y1;
    V "/\" {z} c= V
    proof
A5:   {z} "/\" V = {z "/\" v where v is Element of L: v in V} by YELLOW_4:42;
      let q be object;
      assume q in V "/\" {z};
      then consider v being Element of L such that
A6:   q = z "/\" v and
A7:   v in V by A5;
A8:   {x1} "/\" V = {x1 "/\" s where s is Element of L: s in V} & q = x1
      "/\" (y1 "/\" v) by A6,LATTICE3:16,YELLOW_4:42;
      {y1} "/\" V = {y1 "/\" t where t is Element of L: t in V} by YELLOW_4:42;
      then y1 "/\" v in V "/\" {y1} by A7;
      then q in V "/\" {x1} by A4,A8;
      hence thesis by A2;
    end;
    hence z in G1;
    thus thesis by A1,A3,YELLOW_0:23;
  end;
  hence thesis;
end;
