reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;

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;
