Proof of cancellation law for vector addition

Question

I have a question regarding the proof of the cancellation law for vector addition. This theorem states:

If $x,y,$ and $z$ are vectors in a vector space V such that $x+z=y+z$, then $x=y$.

The proof is as follows:

There exists a vector $v$ in V such that $z+v=0$. Thus

$x=x+0=x+(z+v)=(x+z)+v=(y+z)+v=y+(z+v)=y+0=y$.

My question lies with this equality:

$(x+z)+v=(y+z)+v$.

How can we replace $(x+z)$ with $(y+z)$ and claim to prove anything? Is this equality not contingent on the very thing we are attempting to prove?

Answer 1

It is given that $x+z=y+z$. So, we can definitely use in our proof.

...$\rm V$ such that $x+z=y+z$, then...