# Compactification theorem in Morse Homology

I'm reading "Morse Theory and Floer Homology" by Audin-Damian and got stuck in the proof that the Morse differential indeed induces a homology (i.e. $$\partial^2=0$$). This is basically the compactification theorem of broken trajectories, Section 3.2 in the book. What I can't see is how Proposition 3.2.8 implies Theorem 3.2.7. Here below the details.

Theorem 3.2.7: If Ind(a) = Ind(b) + 2, then $$\bar{\mathcal{L}}(a, b)$$ is a compact manifold of dimension 1 with boundary.

Proposition 3.2.8: Let $$f$$ be a Morse function on a compact manifold $$M$$ and with pseudo-gradient $$X$$ satisfying the Smale condition. Let $$a,c,b\in crit(f)$$ with index $$k-1,k,k+1$$ respectively and $$\lambda_1\in\mathcal{L}(a,c)$$ and $$\lambda_2\in\mathcal{L}(c,b)$$. Then there is a open $$O_{(\lambda_1,\lambda_2)}\subseteq\overline{\mathcal{L}}(a,b)$$ and a continuous injection $$\psi:[0,\delta)\rightarrow O_{(\lambda_1,\lambda_2)}$$ differentiable on the interior and such that \begin{align*} \begin{cases} \psi(0)=(\lambda_1,\lambda_2)\in \overline{\mathcal{L}}(a,b)\\ \psi(s)\in \mathcal{L}(a,b)\ \forall s>0 \end{cases}. \end{align*}

I suppose that this embedding should allow us to show that the dimension of $$\overline{\mathcal{L}}(a,b)$$ is the same as $${\mathcal{L}}(a,b)$$ which we already know, but I can't recall a theorem that states that or see a way to show it.