Corollary 1

— Let E be a Hausdorff weak space and C a proper convex cone of vertex 0 in E that is complete for the uniform structure induced by that of E. Then the mapping $(x, y) \mapsto x + y$ of $C \times C$ in C is proper.

Because of prop. 11, we can suppose that $E = \mathbf{R}^I$ and that $C = \mathbf{R}_+^I$ (GT, I, § 10.1, cor. 1 and 4). But then the mapping $(x, y) \mapsto x + y$ of $C \times C$ in C is written as $((\xi_i), (\eta_i)) \mapsto (\xi_i + \eta_i)$, and we can restrict ourselves to proving that the continuous mapping $f : (\xi, \eta) \mapsto \xi + \eta$ of $\mathbf{R}_+ \times \mathbf{R}_+$ in $\mathbf{R}_+$, is proper (GT, I, § 10.1, cor.3), Now, for all $\zeta \in \mathbf{R}_+$, we see that $f(\zeta)$ is the set of pairs $(\xi, \zeta - \xi)$ such that $0 \leq \xi \leq \zeta$, therefore the inverse image by f of the interval $[0, \zeta]$ is the set of the $(\xi, \eta) \in \mathbf{R}_+ \times \mathbf{R}_+$ such that $\xi + \eta \leq \zeta$, which is compact. The conclusion follows applying (GT, I, § 10.3, prop. 7).