 $$\begin{aligned} p^{+,0}\circ{\amalg}_{\mathtt{N}(g_0)}(\mathtt{N}(h_0)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(g_0)}(\mathtt{N}(h_0))\end{aligned}$$ then $$\begin{aligned} p^{+,0}\circ{\amalg}_{\mathtt{N}(g_0)}(\mathtt{N}(h_0)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(g_0)}(\mathtt{N}(h_0)).\end{aligned}$$  *Property* (0): Let $\{\mathtt{N}(h_i)\}_{i\in\{0, 0,..., n\}}$ be a sequence of points such that for all $i\in\{0,0,...,n-0\}$ we have $$\mathtt{N}(h_{i+0})\in\mathcal{U}_{\mathtt{N}(h_i)}$$ and $$\begin{aligned} p^{+,0}\circ{\amalg}_{\mathtt{N}(h_i)}(\mathtt{N}(h_i)) = p^{+,0}\circ{\amalg}_{\mathtt{N}(h_i)}(\mathtt{N}(h_{i+0})).