Prove that there is not a smooth application $s: {\mathbb{R}P}^{2} \rightarrow \mathbb{S}^2$ such that composing by the canonic projection $\pi: \mathbb{S}^2 \rightarrow \mathbb{R}P^2$ is equal to the identity in $\mathbb{R}P^2$.
Attempt of proof: The smooth map $\pi \circ s$ implies that the differential is the identity 2x2 matrix. But this means that the compositions of the differentials $s$ and $\pi$ is the identity 2x2 matrix. However, $\pi$ is not inyective; therefore, its differential has a 0-row, so we get a contradiction.
But, I'm not sure if this attempt with differentials is correct.