Prove cancellation law using peano axioms.

Question

Using Peano axioms, prove

$∀x∀y∀z(x+y=x+z→y=z)$.

I have been stuck on it for some time, could someone please give a proof? Thanks!

Answer 1

HINT

Since from previous posts I know you are working with a set of Peano axioms that recursively define addition and multiplication over the right operand, it will be much easier to prove the right-cancellation law:

$\forall x \forall y \forall z (y + x = z + x \rightarrow y = z)$

You prove this fairly easily by induction over $x$

So then to prove your original theorem, you would need commutation:

$\forall x \forall y \: x+y = y+x$

which itself can be proven using induction.