Cancellation law for multiplication of natural numbers

Question

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

Answer 1

1) Prove If $x \ne y$ then eithere there exists a $c \ne 0$ so that $x + c =y$ or that $x = y + c$.

2) Prove that if $c \ne 0$ then $z(c + x) \ne z(0 + x)$

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Actually You could use $1$ to define subtraction. If $a \le b$ then, by definition there is a $c$ so that $a + c = b$. If $a \le b$ then define $b - a= c$.

Prove that if $b-a = c$ then $b = a$ if and only if $c = 0$.

Then prove that for all $a,b$ either $a \le b$ or $b \le a$.

And prove subtraction distributes

If $xz = yz$ then $xz \le yz$ and so $0= xz - yz = z(x - y)$. And as $z$ is not $0$ then if $x - y\ne 0$ then $z(x-y) = 0$ we'd get a contradiction. So $x - y = 0$ and $x = y$.

Answer 2

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