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

theorem Th07:
  for X be non empty set,F be Filter of X,
  B be Subset-Family of X st F=<.B.] holds B is basis of F
  proof
    let X be non empty set,
    F be Filter of X,
    B be Subset-Family of X;
    assume
A1: F=<.B.];
    then
A2: B c= F by def3;
    B is non empty
    proof
      assume
A3I1:   B is empty;
A4I2A:  {} c= X;
      {} is Element of B by A3I1,SUBSET_1:def 1;
      then {} in <.B.] by A4I2A,def3;
      hence contradiction by A1,CARD_FIL:def 1;
    end;
    then reconsider B1=B as non empty Subset of F by A2;
    B1 is filter_basis by A1,def3;
    hence thesis;
  end;
