Proof. [deletion-contraction-proof]

$\gdef\quads#1{\quad #1 \quad}$ $\gdef\spaces#1{~ #1 ~}$ $\gdef\without{\setminus}$

$\gdef\str#1{{\footnotesize #1}}$ $\gdef\sstr#1{~{\footnotesize #1}~}$

假定边 $e$ 的两个端点分别为 $A,B$, 不难发现 $$\begin{aligned} \pi(G \without e) &\spaces= \pi(G \without e \sstr{中} A,B ~\str{同色的部分}) ~ + ~ \pi(G \without e \sstr{中} A,B ~\str{异色的部分}) \\ &\spaces= \pi(G / e) ~ + ~ \pi(G) \end{aligned}$$