I have a function, $f(x, y) = (x + y, x)$.
The proof that this function is injective, is as follows:
Say that $f(x,y)=f(x′,y′)$. We are assuming that two different inputs give the same output. For $f$ to be injective we need to prove that the inputs actually are the same. So we have $f(x,y)=f(x′,y′)$ and we need to prove that $x=x′$ and $y=y′$. That $f(x,y)=f(x′,y′)$ means that $(x+y,x)=(x′+y′,x′)$. But if this is true then we certainly have that $x=x′$.
What I don't understand is the "But if this is true" part. If it is true seems to imply that it also could not be true, so how does this proof anything? I mean, the proof seems to be assuming a couple of things, so how does this make for a concrete proof?