$\longmapsto $ }c}
    $u$&$x_2 ^{m} U_2 (x_2)$ \vphantom{$\frac{\frac{1}{1}}{\frac{1}{1}}$} \\
    $v$&$x_2 ^{m+1} V_2 (x_2)$ \vphantom{$\frac{\frac{1}{1}}{\frac{1}{1}}$} 
    \end{tabular} \right. & 
    \begin{tabular} {l} $U_2 (0)$, $V_2 (0) \neq 0 $ \\ $m \geq 1$ \end{tabular}
    \end{array}$$

[*Proof.*]{} Suppose we are given coordinates so that the $f_i$’s are given by (\[bln2\]). Let $b : \tilde X \rightarrow X$ be the blow-up morphism. Let $\tilde u := b^* u$, and note that near the point $\tilde q$ the function $\tilde u$ is a local equation for $E := b^{-1} (q)$, since the tangent vector to the curve locally defined by the vanishing of $u$ is $(0,1)$, while a tangent vector to the curve $C_1$ at $q$ is $(1,0)$. It follows that we may write $b^* v = \tilde u \cdot \tilde v$, and $\tilde u , \tilde v$ is a local system of parameters on $\tilde X$ at $\tilde q$ such that $b$ and its rational inverse $b ^{-1}$ are given by: $$\label{bloco}
b ^* : \left\{ \begin{tabular}{c@{ $\longmapsto $ }c}
$u$& $\tilde u$ \\$v$& $\tilde u \tilde v$ \end{tabular} \right. \hspace{20pt} 
(b ^*) ^{-1} : \left\{ \begin{tabular}{c@{ $\longmapsto $ }c}
$\tilde u$& $u$ \\$\tilde v$& $v / u$ \end{tabular} \right.$$

Thus the morphisms $\tilde f 