Set $M_{K}=\sup_{X}V_{K,\omega}$. If $M_{K}=+\infty$ then $K$ is $PSH(X,\omega)$-polar (theorem 3.2) and there is nothing to prove: $T_{\omega}(K)=Cap_{\omega}(K)=0$. So we assume in the sequel $M_{K}<+\infty$ hence $V_{K,\omega}^{*}\in PSH(X,\omega)$. If $M_{K}\geq 1$ then $u_{K}:=M_{K}^{-1}V_{K,\omega}^{*}\in PSH(X,\omega)$ with $0\leq u_{K}\leq 1$ on $X$. Since $\omega_{V_{K,\omega}^{*}}\leq M_{K}\omega_{u_{K}}$, we get

whence $T_{\omega}(K)\leq\exp(-Cap_{\omega}(K)^{-1/n})$.

If $0\leq M_{K}\leq 1$ then $0\leq V_{K,\omega}^{*}\leq 1$ hence

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