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Let $f: \mathbb R^n \to \mathbb R^m$ be smooth and let $x_0 \in \mathbb R^n$ be such that $\operatorname{rank}{(J_f(x_0))} = k $. Then the constant rank theorem tells us that there exists a neighboudhood of $x_0$ and diffeomorphisms $\phi, \psi$ such that
$$ \phi \circ f \circ \psi = (x_1, \dots,...
