3.  $\varepsilon_2(U_2)\leq \frac{\delta_1(U_1)}3$, whenever $U_i\in \mathscr{U}_i$, $i=1,2$, with $U_1\cap U_2\neq {\varnothing}$.

Since $\mathscr{U}_1$ is locally finite, the paracompact space $X$ has a locally finite open cover $\mathscr{V}$ such that each family $\mathscr{U}_1^V=\left\{U\in \mathscr{U}_1: U\cap \overline{V}\neq
    {\varnothing}\right\}$, $V\in \mathscr{V}$, is finite. For convenience, set $$\label{eq:Near-sel-vgg:1}
    \varepsilon(V)= \min_{U\in
      \mathscr{U}_1^V}\frac{\delta_1(U)}3>0,\quad \text{whenever $V\in
      \mathscr{V}$.}$$ Then by Proposition \[proposition-Finite-C-v4:1\], for each $V\in \mathscr{V}$ there exists an open and locally finite in $\overline{V}$ cover $\mathscr{W}_V$ of the paracompact space $\overline{V}$ and a map $\delta_V:\mathscr{W}_V\to (0,\varepsilon(V))$ such that $\mathscr{W}_V$ refines $\mathscr{U}_1^{V}$ and for every $p,q\in W\in \mathscr{W}_V$, $$\label{eq:Near-sel-vgg:3}
    H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_V(W)}2\quad
    \text{and}\quad \mathbf{O}_{\delta_V(W)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
    \mathbf{O}_{\varepsilon(V)}(\varphi(p)).$$

Finally, we can take $\mathscr{U}_2=\left\{W\cap V: W\in \mathscr{W}_V\ \text{and}\ V\in
    \mathscr{V}\right\}$, which is a locally finite open cover of $X$ because so is $\mathscr{V}$ and each $\mathscr{W}_V$, $V\in \mathscr{V}$, is locally finite in $\overline{V}$. To define the required maps $\delta_2,\varepsilon_2:\mathscr{U}_2\to (0,+\infty)$, for each $U\in \mathscr{U}_2$ fix elements $\lambda(U)\in \mathscr{V}$ and $\mu(U)\in \mathscr{W}_{\lambda(U)}$ with $U=\mu(U)\cap \lambda(U)$, and set $$\label{eq:Near-sel-vgg:4}
    \delta_2(U)=\delta_{\lambda(U)}(\mu(U))\quad \text{and}\quad
    \varepsilon_2(U)= \varepsilon(\lambda(U)),\ \text{
      $U\in \mathscr{U}_2$.}$$

If $U_1\cap U_2\neq {\varnothing}$ for some $U_i\in \mathscr{U}_i$, $i=1,2$, then $U_1\cap \lambda(U_2)\neq {\varnothing}$ and, therefore, $U_1\in \mathscr{U}_1^{\lambda(U_2)}$. According to , we get that $\varepsilon_2(U_2)=\varepsilon(\lambda(U_2)) \leq
  \frac{\delta_1(U_1)}3$. The rest of the properties are evident from and . The proof is complete.

