Proving Uniqueness of Addition in Natural Numbers

[Dansuer]

Homework Statement

Hi everybody, I'm going through the book Fondations of Analysis by E. Landau. I'm trying to prove theorems by myself and then checking if they are correct. But i proved this theorem in a different way then the book and i need a check. thank you

Theorem 4
To every pair of numbers x,y we may assign an unique number x+y such that

1) for every x x+1=x'

2) for every x and y x+y'=(x+y)'

Homework Equations

Axiom 2

for each x there exist an unique number called the successor of x, denoted by x'

Axiom 4

if x'=y' then x=y

The Attempt at a Solution

I proved existence in the same way of the book and so i know it's right.
For uniqueness i did this.

Let's take x,y to be arbitrary natural numbers.
We assume there there exist z and w such that x+y = z and x+y = w and that they satisfies properties 1) and 2).
By axiom 2

(x+y)' = z and (x+y)' = w'

Then by property 2)
x+y' = (x+y)' = z'
= w'

and by axiom 4

z = w

The book prove this by induction, so maybe i missed something.

 

[Dansuer]

Nevermind i see where it's wrong