Given $$T_m(x,y)=\min(x,y),$$ for all $x,y\in[0,1]$.
Prove $T_m\left(x,T_m(y,z)\right)=T_m(T_m(x,y),z)$.
\begin{align*} T_m\left(x,T_m(y,z)\right)&=\min(x,T_m(y,z))\\ &=\min(x,\min(y,z)) \end{align*}
Is it right the proof as below? \begin{align*} T_m\left(x,T_m(y,z)\right)&=\min(x,y,z)\\ &=\min(\min(x,y),z)\\ &=T_m(T_m(x,y),z). \end{align*}