reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;

theorem Th36:
  F c= F "/\" H & H c= F "/\" H
proof
A1: now
    let F,H;
    thus F c= F "/\" H
    proof
      let x be object;
      assume
A2:   x in F;
      then reconsider i = x as Element of L;
      consider p such that
A3:   p in H by SUBSET_1:4;
      i [= i"\/"p by LATTICES:5;
      then
A4:   i"/\"(i"\/"p) = i by LATTICES:4;
      p [= p"\/"i by LATTICES:5;
      then i"\/"p in H by A3,Th9;
      hence thesis by A2,A4;
    end;
  end;
  F "/\" H = H "/\" F by Th33;
  hence thesis by A1;
end;
