[yodsanklai]

A constructive proof would exhibit a solution.

In that case, it has been shown that if a solution exists, it must be that J=6. With this additional information, you can try and find the other digit and conclude the proof (or in the case there's no solution, reach a contradiction).

I don't know how this type of proof is called in English but it's quite common. First come up with necessary conditions on your potential solution, and then you use these conditions to build an actual solution.