How can we show that the multi-polylogarithmic function $L_{\underbrace{1,\ldots,1}_n}(z)=\frac{1}{n!}(L_1(z))^n$

Question

How can we show that the multi-polylogarithmic function $$L_{\underbrace{1,\ldots,1}_n}(z)=\frac{1}{n!}(L_1(z))^n.$$ Here $L_1(z)=-log(1-z)$.

I know that $\frac{d}{dz}L_{k_1,\ldots,k_r}(z)=\frac{1}{1-z}L_{k_1,\ldots,k_{r-1}}(z)$ for $k_r=1$. Please help me with this.