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

theorem Th13:
  for X,Y be non empty set,f be Function of X,Y,
  F be Filter of X holds f.:F is filter_base of Y &
  <.f.:F.]=filter_image(f,F)
  proof
    let X,Y be non empty set,f be Function of X,Y,F be Filter of X;
    reconsider F1=F as basis of F by Th03;
    f.:( #F1 ) is filter_base of Y &
    <.f.:( #F1 ).]=filter_image(f,F) by Th12;
    hence thesis;
  end;
