Cohomology ring of Grassmannians

I’m reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing):

Let $w=1+w_1+ \ldots + w_m$ be the total Stiefel-Whitney class of the canonical $m$-plane bundle over $G_m(\mathbb{R}^{m+n})$ and let $\bar{w}=1+\bar{w_1}+\ldots+ \bar{w_n}$ be its dual. Then $H^\ast G_m (\mathbb{R}^{m+n})$ is the quotient of the polynomial algebra $\mathbb{Z}_2[1,w_1,\ldots,w_m]$ by the ideal $(\bar w_{m+1},\cdots,\bar {w}_{m+n})$ generated by the relation $w\bar{w}=1$.

The reference provided is to Borel’s La cohomolgie mod 2 de espaces homogenes. As the title suggests, this paper is in French, a language with which I am not familiar.

I’m familiar with the fact that the cohomology ring of the infinite Grassmannian $G_m(\mathbb{R}^\infty)$ is freely generated by $w_1,\ldots,w_m$ over $\mathbb{Z}_2$ (as proved in Hatcher’s Vector Bundles), but I can’t see how to prove this variant. Any help would be much appreciated. Perhaps someone can even translate the proof given in Borel’s paper.

Solutions Collecting From Web of "Cohomology ring of Grassmannians"