Weak partial derivatives on a planar domain can be tested against $C^1$ functions

Let $E$ be a finite-dimensional real normed space and let $U\subseteq\mathbb{C}$ be open. Suppose $u_s,u_t:\mathbb{C}\to E$ are weak partial derivatives of $u:\mathbb{C}\to E$ on $U$, i.e. $\int(\partial_s\varphi)\,u=-\int\varphi\,u_s$ and $\int(\partial_t\varphi)\,u=-\int\varphi\,u_t$ for every real $C^\infty$ function $\varphi$ with compact support in $U$, and suppose $u,u_s,u_t$ are integrable on every compact subset of $U$. Then the same two identities hold for every real $C^1$ function $\varphi$ with compact support in $U$.

This lets weak partial derivatives be tested against $C^1$ functions, for instance products of a test function with a $C^1$ coefficient.

Formalization Note The hypothesis is HasWeakPartialsOn U u us ut together with IntegrableOn of u, us, ut on every compact K ⊆ U. For the test function: ContDiff ℝ 1 φ, HasCompactSupport φ, tsupport φ ⊆ U; $\partial_s\varphi(z)$ is fderiv ℝ φ z 1 and $\partial_t\varphi(z)$ is fderiv ℝ φ z Complex.I. The integrals are Bochner integrals over $\mathbb{C}$.