Proposition 5

The hypotheses and notations being those of def. 7, the linear mappings $s_\Phi$ and $d_\Phi$ associated with $\Phi$ have the same rank, and this rank is equal to the rank of the form $\Phi$.

Indeed the kernel of the mapping $d_\Phi$ from F into $E^*$ is obviously $E^0$, hence its rank is equal to the dimension of $F/E^0$. Analogously the rank of $s_\Phi$ is equal to the dimension of $E/F^0$.