In a proof of the "Degree 0 Lemma" i.e.
Lemma : Let $\tilde{M}$ be a compact $(n+1)-$manifold, $M = \partial \tilde{M}$ with induced orientation, $M$ compact and $f : M \longmapsto N$ of class $C^{\infty}$ with dim$M$=dim$N$. If $f$ extends to some $F : \tilde{M} \longmapsto N$ then deg($f,y) = 0$, for all $y \in \hspace{0.1cm} $RegVal($f$).
Where deg($f,y) = \sum\limits_{x \in f^{-1}(y)}\text{sgn}(df_x) \hspace{0.1cm} \forall \hspace{0.1cm} y \in \hspace{0.1cm} $RegVal($f$), at a certain point, in particular after defined an orientation of an arc $A$ of the manifold $F^{-1}(y)$, the idea woulde be to carry around a basis from the starting point of $\partial A \in M$ to end point point of the same and get cancellation of the signs.
In order to do so, we shift the attention to the following Lemma, which I'm unable to prove since I think it's related to the rank theorem or canonical form which has $f$ under appropriate charts (theorem which has not been proven in my course so I don't even see how could be easily follow from that) :
Lemma : Let $\tilde{M}$ be an oriented manifold with boundary and $\gamma : [0,1] \longmapsto \tilde{M}$ regular curve. Then $\forall \hspace{0.1cm} t_0 \in [0,1]\hspace{0.1cm} \exists \hspace{0.1cm} \epsilon > 0 : \forall \hspace{0.1cm} v_2,\cdots,v_n \in T_{\gamma(t_0)}\tilde{M}$ such that $\gamma'(t_0),v_2,\cdots,v_n$ basis of $T_{\gamma(t_0)}\tilde{M}$ then exist smooth tangent vectors fields along $\gamma$, $X_i : (t_0-\epsilon,t_0+\epsilon) \cap [0,1] \longmapsto T_{\gamma(t_0)}\tilde{M}$ such that $\begin{cases} X_i(t_0) = v_i \hspace{0.1cm} i=2,\cdots,n\\ \gamma'(t),X_2(\gamma(t)),\cdots,X_n(\gamma(t)) \hspace{0.1cm} \text{basis of} \hspace{0.1cm} T_{\gamma(t)}M \end{cases} \forall \hspace{0.1cm} t \in (t_0-\epsilon,t_0+\epsilon) \cap [0,1]$
There is a proof of the theorem which doesn't rely on the thoerems of ranks cited above ? I'd like to have a "real" proof of this important Lemma whithout involving unproven theorems if possible. In any case, any reference or direct proof (even with rank theorem) would be appreciated.