Para mostrar que $W_k^n(m,j)\subset W_{k+1}^n(m,j)$ seja $x\in W_k^n(m,j)$ então existe $f\in A_{j,k}^n$ tal que $x\in V_{f(1)}(m,f(0)),\ldots,V_{f(\ell-1)}(m+\ell-2,f(\ell-2))$ onde $dom(f)=\ell$ e $f(\ell-1)=k$. Defina $g:\ell\rightarrow\omega$ tal que $g(a)=f(a)$, $a<\ell-1$ e $g(\ell-1)=k+1$.