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

theorem Th13bThmBA2:
  for X,Y be non empty set, f be Function of X,Y,
  F be Filter of X, B be basis of F holds
  f.:#B is basis of filter_image(f,F) &
  <.f.:( #B ).] = filter_image(f,F)
  proof
    let X,Y be non empty set,f be Function of X,Y,
    F be Filter of X,B be basis of F;
    reconsider FB=f.:( #B ) as filter_base of Y by Th12;
    reconsider FB2=<.FB.) as Filter of Y;
    FB2=filter_image(f,F) by Th12;
    hence thesis by Th10;
  end;
