Provide reasons (mostly vector space properties) as justification for each of the seven steps of the following proof.

Theorem For any vectors $\vect{u},\,\vect{v},\,\vect{w}\in\complex{m}$, if $\vect{u} + \vect{v} = \vect{u} + \vect{w}$, then $\vect{v} = \vect{w}$.

Proof: Let $\vect{u},\,\vect{v},\,\vect{w}\in\complex{m}$, and suppose $\vect{u} + \vect{v} = \vect{u} + \vect{w}$. \begin{align*} \vect{v} &= \zerovector + \vect{v}\\ &= (-\vect{u} + \vect{u}) + \vect{v}\\ &= -\vect{u} + (\vect{u} + \vect{v})\\ &= -\vect{u} + (\vect{u} + \vect{w})\\ &= (-\vect{u} + \vect{u}) + \vect{w}\\ &= \zerovector + \vect{w}\\ &= \vect{w} \end{align*}