Prove the assumption of uniqueness is not necessary

[Kinetica]

Is my proof correct?

Homework Statement

Show that, if there exists a number 0 for which x+0=x for all x∈R, and a number 0' for which x+0'=x for all x∈R, then 0=0'.

The Attempt at a Solution

Proof by contradiction:

Assume, 0≠0'. Then,

x+0=x and x+0'=x'

Such that:

x=x-0 and x=x'-0'

Such that: x-0=x'-0'

Which is a contradiction.
Thus, 0=0'.

 

[micromass]

I don't see the contradiction...
And what is x'??