Let $M$ be a smooth manifold, let $n \in \mathbb{N}$, and let
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\begin{align*}
P: M \to M_n(\mathbb{R})
\end{align*}
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be a smooth map such that $P(p)^2 = P(p)$ for every $p \in M$. Suppose that there exists $k \in \{0,1,\dots,n\}$ such that $\operatorname{rank} P(p) = k$ for every $p \in M$. Define