your inequality is not correct. it should be \(\displaystyle \geq\) instead of \(\displaystyle \leq.\) let \(\displaystyle T^{-1}(Z)=X\) and define the map \(\displaystyle S: V \longrightarrow W/Z\) by \(\displaystyle S(v)=T(v) + Z.\) clearly \(\displaystyle \ker S = X\) and thus \(\displaystyle V/X \cong S(V)=T(V)/Z.\)

hence \(\displaystyle \dim V - \dim X = \dim V/X = \dim T(V)/Z = \dim T(V) - \dim Z \leq \dim W - \dim Z.\) therefore \(\displaystyle \dim X \geq \dim V - \dim W + \dim Z.\)