Yes. Marcher literally means to walk, but it also very commonly means to work, to function, or to be effective.
Examples:
So in this sentence, marcher does not mean physical walking.
Both can mean it works, but they are not exactly the same in tone.
For people, plans, arguments, methods, and everyday situations, French often prefers ça marche.
So Avec cette preuve, ça marche sounds natural in ordinary French.
It refers to something understood from the context.
For example, it could mean:
So the sentence does not mean that the proof itself is literally working like a machine. It means something like:
French often uses ça when the exact subject is understood from context.
The comma is helpful, but not always absolutely required.
It marks a pause after the introductory phrase Avec cette preuve.
So:
Both can be seen, but the version with the comma is clearer and more natural in writing because it separates the setup from the main clause.
Sometimes, but the nuance changes.
So:
If you do not want to emphasize gratitude or a strongly positive cause, avec is safer.