theorem Th1:
  F1 /\ F2 is Filter of L
proof
  consider p such that
A1: p in F1 by SUBSET_1:4;
  consider q such that
A2: q in F2 by SUBSET_1:4;
A3: p "\/" q in F2 by A2,FILTER_0:10;
  p "\/" q in F1 by A1,FILTER_0:10;
  then reconsider D = F1 /\ F2 as non empty Subset of L by A3,XBOOLE_0:def 4;
  now
    let p,q;
    p "/\" q in F1 & p "/\" q in F2 iff p in F1 & q in F1 & p in F2 & q in
    F2 by FILTER_0:8;
    hence p in F1 /\ F2 & q in F1 /\ F2 iff
      p "/\" q in F1 /\ F2 by XBOOLE_0:def 4;
  end;
  then D is Filter of L by FILTER_0:8;
  hence thesis;
end;
