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

theorem
  for X be non empty set,B1,B2 be filter_base of X st <.B1.)=<.B2.) holds
  B1,B2 are_equivalent_generators
  proof
    let X be non empty set,B1,B2 be filter_base of X;
    assume
A1: <.B1.)=<.B2.);
A2: for b1 be Element of B1 ex b2 be Element of B2 st b2 c= b1
    proof
      let b1 be Element of B1;
      b1 in <.B2.] by A1,def3;
      then consider b2 be Element of B2 such that
A3:   b2 c= b1 by def3;
      thus thesis by A3;
    end;
    for b2 be Element of B2 ex b1 be Element of B1 st b1 c= b2
    proof
      let b2 be Element of B2;
      b2 in <.B1.] by A1,def3;
      then consider b1 be Element of B1 such that
A4:   b1 c= b2 by def3;
      thus thesis by A4;
    end;
    hence thesis by A2;
  end;
