Let $A$ be an abelian scheme over a base scheme $S$ and $\omega$ a global section of the differential module $\Omega^1_{A\times_S A/S}$.

Suppose that $\omega$ is zero when restricted to $A\times S$ and $S\times A$, both times via the zero section and the identity.

Then why can one conclude that $\omega$ itself is already zero?