A confustion about the proof of Primary Decomposition theorem In proofWiki, in the proof of Primary Decomposition Theorem while showing that (in the induction step)
$$\ker \left({p_i \left({T \restriction_W}\right)^{a_i} }\right) = \ker \left({p_i \left({T}\right)^{a_i} }\right),$$
how does the author concludes 
$$p_i \left({T \restriction_W}\right)^{a_i} \left({v}\right) = p_i \left({T}\right)^{a_i} \left({v}\right) = 0$$
for $v \in \ker \left({p_j \left({T}\right)^{a_j} }\right)$ ?
Edit:
I'm referring to the proof that I have linked.

Since $v \in \ker \left({p_j \left({T}\right)^{a_j} }\right)$ implies $v \in W$, and when the input from the space $W$, both $T|_{W}$ and $T$ are the same, we should have 
$$p_i \left({T \restriction_W}\right)^{a_i} \left({v}\right) = p_i \left({T}\right)^{a_i} \left({v}\right) = 0$$
