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

theorem
  for X be non empty set,F1,F2 be Filter of X, F be Filter of X st
  F is_filter-finer_than F1 & F is_filter-finer_than F2 holds
  for M1 be Element of F1,M2 be Element of F2 holds M1/\M2 is non empty
  proof
    let X be non empty set,F1,F2 be Filter of X, F be Filter of X such that
A1: F is_filter-finer_than F1 and
A2: F is_filter-finer_than F2;
    hereby
      let M1 be Element of F1,M2 be Element of F2;
      M1 in F1 & M2 in F2;
      then M1/\M2 in F by A1,A2,CARD_FIL:def 1;
      hence M1/\M2 is non empty by CARD_FIL:def 1;
    end;
  end;
