theorem Th40:
  for f being Function of X,Y, cFXa,cFXb being Filter of X st
  cFXb is_filter-finer_than cFXa holds
  filter_image(f,cFXb) is_filter-finer_than filter_image(f,cFXa)
  proof
    let f be Function of X,Y, cFXa,cFXb be Filter of X;
    assume
A1: cFXb is_filter-finer_than cFXa;
    filter_image(f,cFXa) c= filter_image(f,cFXb)
    proof
      let x be object;
      assume x in filter_image(f,cFXa);
      then ex M be Subset of Y st x = M & f"(M) in cFXa;
      hence thesis by A1;
    end;
    hence thesis;
  end;
