Cancellation law for multiplication of natural numbers I'm trying to prove the following cancellation law for multiplication of natural numbers: if $xz=yz$ for natural numbers $x,y$ and $z$, where $z$ is non-zero, then $x=y$.
I'm working with the peano-axioms and I've already proven elementary properties of multiplication such as commutativity, left and right-distributivity etc.
I'm thinking that the proof could maybe be done by using induction on one of the variables while the other two are fixed, but I'm not sure where to start.
Thanks

I would write
$$xz-yz=0$$ so we get $$z(x-y)=0$$ since $z\neq 0$ we obtain $$x=y$$
