Let $f:\Bbb{R}\rightarrow\Bbb{R}$ a function of class $C^{1}$ such that $|f'(t)|\leq k < 1\, \forall\, t\in\Bbb{R}$. Define $\phi:\Bbb{R}^{2}\rightarrow\Bbb{R}^2$ by
$$\phi(x,y)=(x+f(y),y+f(x)). $$
I need to show that $\phi$ is a diffeomorphism.
I want to use this theorem:
$\textbf{Theorem:}$ Let $A\subset\Bbb{R}^{n}$ an open set, and $\phi:A\rightarrow\Bbb{R}^{n}$ a function of class $C^{r}.$ If $\phi$ is injective and $\phi'(\textbf{x})$ is non-singular $\forall\, \textbf{x}\in A$, so $\phi$ is a diffeomorphism of class $C^{r}.$
In this case, $A=\Bbb{R}^{2}$ itself, and $r=1$. It was easy to show that $\phi'(x)$ is non-singular for all $\textbf{x}\in\Bbb{R}^{2},$ but I don't know how to show that $\phi$ is injetive.
What I did:
$$\phi(x_1,y_1)=\phi(x_2,y_2)\iff \begin{cases} x_1+f(y_1)=x_2+f(y_2) \\ y_1+f(x_1)=y_2+f(x_2) \end{cases} \iff \begin{cases} x_1-x_2=f(y_2)-f(y_1) \\ y_1-y_2=f(x_2)-f(x_2) \end{cases}. $$
If I proved that this two equations are both zero, I 'm done, but I don't know if this is the right way.