Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Deduce that $A$ is $F$-regular for every non-zero vector subspace $F$ of $E_{n}$.
Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Deduce that $A$ is $F$-regular for every non-zero vector subspace $F$ of $E_{n}$.