Smooth even Hadamard factorisation of the odd part

Let $E$ be a finite-dimensional real normed space and $F$ a complete real normed space, and let $B : E \times \mathbb{R} \to F$ be $C^\infty$ (ContDiff ℝ ⊤). Then there exists a map $Q : E \times \mathbb{R} \to F$ which is again $C^\infty$, which is even in its last variable in the sense that $Q(e,-\rho) = Q(e,\rho)$ for all $e \in E$ and all $\rho \in \mathbb{R}$, and which satisfies

for all $e \in E$ and $\rho \in \mathbb{R}$, the scalar $2\rho$ acting on $F$ by its real scalar multiplication. Thus the odd part of $B$ in the last variable factors as $\rho$ times a globally smooth function that is even in $\rho$; no compact support or decay assumption is imposed on $B$, and the statement is an existence assertion, with no uniqueness or explicit formula for $Q$ claimed.

This is Hadamard's division lemma with parameters, in the form adapted to reflection in the last variable: the odd part of a smooth function vanishes to first order on $\rho = 0$ and the quotient may be chosen smooth and even. It is used in the even-reflection step that splits a smooth function of $|\rho|$ into a smooth even part plus $|\rho|$ times a smooth factor, and is cited by MeasureTheory.exists_contDiff_integral_mul_log_sq_add_sq_eq_add_abs_mul_of_hasCompactSupport.