Full column rank of the sensitivity Jacobian implies local identifiability

Full column rank means the $p$ columns of $J_{\xi }({\theta }_0)$ are independent. They lie in ${\mathbb{R}}^m$, so [F3] gives $p\le m$. By [F2] the matrix rank is $p$, and [F4] supplies a set $I$ of $p$ rows whose square submatrix $B=J_{\xi }({\theta }_0)[I,:]$ has nonzero determinant.

By [F5], $B$ is invertible. Let $P_I:{\mathbb{R}}^m\to {\mathbb{R}}^p$ be the coordinate projection onto the rows in $I$, and set $G=P_I\circ F_{\xi }$. The linear map $P_I$ has derivative $P_I$, since its linear first-order expansion has zero remainder. Thus [F1] and [F6] give $DG(\theta )=P_I{J}_{\xi }(\theta )$ for every $\theta \in \mathrm{\Theta }$; this derivative is continuous, so $G$ is $C^1$, and $DG({\theta }_0)=P_I{J}_{\xi }({\theta }_0)=B$.

Apply [F7] to $G$: there is an open neighborhood $V\subseteq \mathrm{\Theta }$ of ${\theta }_0$ on which $G$ is injective. If $\theta ,{\theta }^{\mathrm{\prime }}\in V$ satisfy $F_{\xi }(\theta )=F_{\xi }({\theta }^{\mathrm{\prime }})$, then applying $P_I$ gives $G(\theta )=G({\theta }^{\mathrm{\prime }})$, hence $\theta ={\theta }^{\mathrm{\prime }}$. Thus $F_{\xi }$ is injective on $V$, which is local identifiability by [F1]. The argument selects one of finitely many row minors and uses no choice axiom.