Corollary 2

The hypotheses being those of cor. 1, let $M, N$ be two subspaces of $E$; then one has $(M + N)^0 = M^0 \cap N^0$; if moreover $M$ and $N$ are of finite dimension, one has $(M \cap N)^0 = M^0 + N^0$.

The first assertion is trivial. Suppose that M and N are of finite dimension, and let G = M^0 + N^0; one has G^0 = M^{00} \cap N^{00} = M \cap N according to cor. 1; the prop. 4 applied to the restriction of $\Phi$ to $M \times G$ then shows (since $M^0 \subset G$ and $G^0 \subset M$) that one has $\dim M/(M \cap N) = \dim G/M^0 = \operatorname{codim} M^0 - \operatorname{codim} G$, and since $\operatorname{codim} M^0 = \dim M$, one deduces $\dim (M \cap N) = \operatorname{codim} G$. But one also has $\dim (M \cap N) = \operatorname{codim} (M \cap N)^0$ according to cor. 1, and since $G \subset G^{00} = (M \cap N)^0$ one has $G = (M \cap N)^0$.

Prop. 4 permits one to give the following definition: