reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem
  for X1,X2 being non empty set,F1 being Filter of X1,F2 being Filter of X2
  holds
  the set of all [:f1,f2:] where f1 is Element of F1,f2 is Element of F2 is
  non empty Subset-Family of [:X1,X2:]
  proof
    let X1,X2 be non empty set,F1 be Filter of X1,F2 be Filter of X2;
    set F1xF2=the set of all [:f1,f2:] where f1 is Element of F1,
    f2 is Element of F2;
A0: [:the Element of F1,the Element of F2:] in F1xF2;
    F1xF2 c= bool [:X1,X2:]
    proof
      let x be object;assume x in F1xF2;
      then consider f1 be Element of F1,f2 be Element of F2 such that
A1:   x=[:f1,f2:];
      thus thesis by A1;
    end;
    hence thesis by A0;
  end;
