 Since "Y""k" − {"p"} deformation retracts onto the subspace "Y""k"-"e""k", we can further homotope the restriction of "f" to formula_4 to a map, say, "g", with the property that "g"("e""n") misses the cell "e""k" of "Y", still relative to "X""n-1".Since "f"("e""n") met only finitely many cells of "Y" to begin with, we can repeat this process finitely many times to make formula_6 miss all cells of "Y" of dimension larger than "n".We repeat this process for every "n"-cell of "X", fixing cells of the subcomplex "A" on which "f" is already cellular, and we thus obtain a homotopy (relative to the ("n" − 1)-skeleton of "X" and the "n"-cells of "A") of the restriction of "f" to "X""n" to a map cellular on all cells of "X" of dimension at most "n".