Morse-Sard for smooth manifolds

If $\dim N=0$, then every differential $d{F}_p:T_{p}M\to T_{F(p)}N=\{0\}$ is surjective, so [F1] makes every value of $F$ regular. Thus the critical value set is empty, which is null by [F2]. Assume henceforth that $\dim N>0$.

By [L3] under Countable Choice, take countable smooth atlases $\{(U_i,{\phi }_i)\}$ on $M$ and $\{(V_j,{\psi }_j)\}$ on $N$; the latter detects nullity by [L1]. The countable open overlaps $U_{ij}=U_i\cap F^{-1}(V_j)$ cover $M$.

For each pair $(i,j)$, the coordinate representative $f_{ij}:={\psi }_j\circ F\circ {\phi }_i^{-1}$ on ${\phi }_i(U_{ij})$ is smooth between Euclidean open sets with positive-dimensional target. A point of $U_{ij}$ is critical for $F$ exactly when its coordinate representative is critical for $f_{ij}$, because the chart maps have invertible differentials. By [L2], ${\psi }_j(F(\mathrm{Crit}(F)\cap U_{ij}))$ is null.

Fix $j$. The chart image of the critical value set inside $V_j$ is the countable union over $i$ of the null sets from step 2.1. By the countable-union clause of [L1] in Euclidean coordinates, this chart image is null. Since this holds for every $j$, [L1] detects the critical value set of $F$ as null in $N$.