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

theorem Th08:
  for X be non empty set,B be filter_base of X holds <.B.] is Filter of X
  proof
    let X be non empty set, B be filter_base of X;
    consider b0 be object such that
A1: b0 in B by XBOOLE_0:def 1;
    now
      thus <.B.] is non empty by A1,def3;
      hereby
        assume
A2:     {} in <.B.];
        then reconsider x0={} as Subset of X;
        consider b0 be Element of B such that
A3:     b0 c= x0 by A2,def3;
        thus not {} in <.B.] by A3;
      end;
      let Y1,Y2 be Subset of X;
      hereby
        assume that
A4:     Y1 in <.B.] and
A5:     Y2 in <.B.];
        reconsider y1=Y1,y2=Y2 as Subset of X;
        consider b1 be Element of B such that
A6:     b1 c= y1 by A4,def3;
        consider b2 be Element of B such that
A7:     b2 c= y2 by A5,def3;
        consider b3 be Element of B such that
A8:     b3 c= b1/\b2 by def4;
        b1/\b2 c= Y1/\Y2 by A6,A7,XBOOLE_1:27;
        then b3 c= Y1/\Y2 by A8;
        hence Y1/\Y2 in <.B.] by def3;
      end;
      assume that
A9:   Y1 in <.B.] and
A10:  Y1 c= Y2;
      reconsider y1=Y1 as Subset of X;
      consider b1 be Element of B such that
A11:  b1 c= y1 by A9,def3;
      b1 c= Y2 by A10,A11;
      hence Y2 in <.B.] by def3;
    end;
    hence thesis by CARD_FIL:def 1;
  end;
