[lmkg]

Excellent walk-through, thank you. You also removed one of my uses of additive inverses, which gets us closer to not needing the negative integers. I have two ideas for getting rid of the last use of them in your last step, but I'm sure they're kosher.

1) From the equation x0 + x0 = x0, you don't need inverses, just cancellation. I believe that cancellation is a strictly weaker property.

2) From the equation x0 + x0 = x0, note that x0 is the additive identity. Inverses are unique, thus x0 = 0.

Unfortunately, I'm not familiar enough with Peano arithmetic to know if proving either of these statements requires the statement we're trying to prove, that x*0 = 0. I'm more familiar with algebra, where inverses exist axiomatically. But at least we've weakened the hypotheses!

[jules]

To prove something about multiplication you first need to provide a definition of multiplication. So what is you definition of multiplication? The usual definition is:

    a*0 = 0
    a*S(b) = a*b+a

Cancellation follows from injectivity of S by induction.

[python123]

No, you can't say x0 is the additive identity because 0 is the only additive identity. That would be equivalent to x0=0. Your way is fine if you just use what archgoon wrote to complete it.