Compact Lie-group actions are proper

Let $K\subseteq M\times M$ be compact and let $D={\mathrm{pr}}_2(K)$. The projection is continuous, so $D$ is compact by [F2].

Since both coordinates of every $(g,x)\in {\mathrm{\Theta }}^{-1}(K)$ satisfy $(g\cdot x,x)\in K$, its second coordinate $x$ lies in $D$. Hence ${\mathrm{\Theta }}^{-1}(K)\subseteq G\times D$, and $G\times D$ is compact by [F3].

The manifold $M$ is Hausdorff by hypothesis, so $M\times M$ is Hausdorff by [F5]. Thus $K$ is closed by [F4]. The map $\mathrm{\Theta }$ is continuous because the action and the second projection are continuous, so ${\mathrm{\Theta }}^{-1}(K)$ is closed in $G\times M$, and therefore also closed in the subspace $G\times D$.

By [F4], the closed subset ${\mathrm{\Theta }}^{-1}(K)$ of the compact space $G\times D$ is compact. Since $K$ was arbitrary, [F1] proves properness. Local compactness of $M$ is part of the stated manifold context but is not needed in this compact-domain argument; no freeness or choice principle is used.