\varphi({\overrightarrow{C}})= e^{i\langle  I(\overrightarrow{C}),  \theta(\Phi)\rangle}$, where $ \langle,\rangle $ denotes the usual inner product. Let $ \theta(\Phi_{1}) =(\alpha_{1}, \alpha_{2}, \dots, \alpha_{m})$ and $ \theta(\Phi_{2}) =(\beta_{1}, \beta_{2}, \dots, \beta_{m})$. If $ \overrightarrow{C}$ is any directed cycle, then, $$\begin{aligned}
    \varphi_{1}(\overrightarrow{C}) &=& e^{i\langle  I(\overrightarrow{C)}, \theta(\Phi_{1})\rangle} \\
        &=& e^{i \langle c_{1}I(\overrightarrow{C_{1}(T)})+c_{2}I(\overrightarrow{C_{2}(T)})+ \dots+ c_{m-n+1}I(\overrightarrow{C_{m-n+1}(T)}), \theta(\Phi_{1}) \rangle }\\        
    &=& e^{{ c_{1} i \langle I(\overrightarrow{C_{1}(T)}) , \theta(\Phi_{1}) \rangle }+{c_{2} i \langle I(\overrightarrow{C_{2}(T)}) , \theta(\Phi_{1}) \rangle }+ \dots +{c_{m-n+1} i\langle  I(\overrightarrow{C_{m-n+1}(T)}) , \theta(\Phi_{1})\rangle }}\\
        &=&  \left({e^{i \langle  I(\overrightarrow{C_{1}(T)}) , \theta(\Phi_{1}) \rangle  }} \right) ^{c_{1}} \left( {e^{i  \langle I(\overrightarrow{C_{2}(T)}) , \theta(\Phi_{1}) \rangle }}\right)^{c_{2}}\dots \left( {e^{i \langle  I(\overrightarrow{C_{m-n+1}(T)}) , \theta(\Phi_{1}) \rangle  }}\right) ^{c_{m-n+1}}  \\
        &=&  \left({e^{i \langle  I(\overrightarrow{C_{1}(T)}) , \theta(\Phi_{2}) \rangle  }} \right) ^{c_{1}} \left( {e^{i  \langle I(\overrightarrow{C_{2}(T)}) , \theta(\Phi_{2}) \rangle }}\right)^{c_{2}}\dots \left( {e^{i \langle  I(\overrightarrow{C_{m-n+1}(T)}) , \theta(\Phi_{2}) \rangle  }}\right) ^{c_{m-n+1}}  \\
        &=& \varphi_{2}(\overrightarrow{C}).
    \end{aligned}$$

Let $ \varphi_{1} $ and $ \varphi_{2} $ be two $ \mathbb{T}$-gains on a connected graph $ G $ with $ n $ vertices and $ m $ edges. If $ \varphi_{1}( \overrightarrow{C_{k}(T)})=\varphi_{2}( \overrightarrow{C_{k}(T)}) $ for $ k=1, 2, \dots, (m-n+1) $ holds for a normal spanning tree $ T $, then this holds for any normal spanning tree of $ G $.

Therefore, in corollary 