I have been reading some notes on Morse Theory and at some point the following claim is made:
Suppose we have $f$ a Morse function and $M$ a compact smooth manifold, with $x,y\in Crit(f)$ such that $m(x)=m(y)+1$. Assume that we have transversality. If $q\in W^{u}(q_1)$ then $q\in W^{u}(q_0)$ and moreover there exists a neighborhood $U$ of $q$ in $M$ and a homeomorphism $\Phi:[0,\infty[\times \mathbb{R}^{k}\rightarrow U\cap \overline{W^{u}(q_0)}$ such that
$$ \phi(0,0)=q$$ $$\forall x\in [0,\infty[, x_0\mapsto \phi(x_0,\cdot) \hspace{2mm} \text{is a diffeomorphism}$$ $$\phi(0,\cdot) \hspace{2mm}\text{is a diffeomorphism onto a neighborhood of $q$ in $W^{u}(q_1)$}$$
$$\phi((0,\infty)\times \mathbb{R}^{k})\subset W^{u}(q_0)$$
The author claims that this following from the properties of the graph transform near the hyperbolic fixed point $q_1$ and due to the fact that $W^{u}(q_0)\pitchfork W^{s}(q_1)$. However, I am not being able to prove this result using the properties of the graph transform. Does anyone have any idea on how to prove this, or knows of any reference that deals with this ?
Any insight is appreciated, thanks in advance.