Why is $(T + N_X)(x) \subset T(x)$ when $Dom T \subset X$?

Question

I'm trying to show that given a maximal monotone operator $T$ and a closed convex set $X$ with $Dom T \subset X$ then for a given $x \in Dom T$ it holds

$(T + N_X)(x) \subset T(x)$

where $N_X(\cdot)$ is the normal cone, that is

$N_X(x) = \{y\text{ st }\langle y, x - u\rangle \geq 0 \quad \forall u \in X\}$

But can't seem to figure it out. Any ideas?

Answer 1

Note that $T\subset T+N_X$ (because $Dom(T)\subset X$ and $0\in N_X (x)$, $\forall x\in X$) and because $T$ is maximal while $T+N_X$ is monotone one gets the equality $T=T+N_X$. You do not need $X$ closed convex.