.  Let $Z$ be in $\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{A}$ such that $Z = \mathtt{N}(g)$ for some $g\in\mathsf{U}_{\hbox{\tiny $\mathrm{rec}$}}\mathbb{H}$ and $W\in{\mathcal{L}}_Z^{+}$. Therefore there exist real numbers $s_0$, $s_0$ such that $$W = (\tilde{{N}}(g) + s_0 \nu^+(g), \nu(g) + s_0 \nu^+(g)).$$ 