Why does the following theorem both provide a neccesary and sufficient condition for a $2 \times 2$ linear system of congruences to have a unique solution ?
I see that $\gcd((ad-bc),m) = 1$ is a neccesary condition for the solutions to exists.
However for this condition to imply that there exists unique solutions $x \equiv x_0 (\mod m)$ and $y \equiv y_0 (\mod m)$ to the original system, we must work backwards ? But how is this possible ?
In the proof we actually assume the solutions exist and then find the neccesary condition, namely $\gcd((ad-bc),m) = 1$. However, the other way is unclear to me?