How to proof an injective(one-to-one) function

Let $f:A\rightarrow{A}$ with $f(x)=\frac{x+2}{x-3}$ for $A=\mathbb{R}/\{1,3\}$

Show that $f$ is a one-to-one function.

Proof: we are given $f:A\rightarrow{A}$ with $f(x)=\frac{x+2}{x-3}$ and we are to proof that $f$ is a one-to-one function. Recall that a one-to-one function is defined thus: $\forall{x,y}\in{A}(f(x)=f(y)\Rightarrow{x=y})$.

Let $x,y\in{A}$ with $f(x)=f(y)$

$\Rightarrow\frac{x+2}{x-3}=\frac{y+2}{y-3}$

$\Rightarrow(x+2)(y-3)=(x-3)(y+2)$

$\Rightarrow{xy-3x+2y-6=xy+2x-3y-6}$

$5y=5x\Rightarrow{y=x}$

Hence, $y=x$ satisfies our definition of a one-to-one function.