Avec cette preuve, ça marche.

Because preuve is a feminine singular noun in French.

The demonstrative adjective this/that must agree with the noun:

• ce = masculine singular
• cette = feminine singular
• ces = plural

So:

• ce livre = this book
• cette preuve = this proof

Avec usually means with.

In Avec cette preuve, ça marche, it means something like:

• with this proof
• using this proof
• given this proof

So the idea is that this proof makes something work, supports it, or makes it valid.

Ça is an informal, very common word meaning that or it.

It is the spoken and everyday equivalent of cela.

So:

• ça marche = it works / that works
• cela marche = same meaning, but more formal or careful

In everyday French, ça is extremely common.

The cedilla under ç makes the c sound like s before a, o, or u.

So:

• ça is pronounced roughly like sa
• without the cedilla, ca would normally sound with a hard k sound

The cedilla is there only to control pronunciation.

Yes. Marcher literally means to walk, but it also very commonly means to work, to function, or to be effective.

Examples:

• Je marche dans la rue. = I’m walking in the street.
• Mon téléphone ne marche pas. = My phone isn’t working.
• Ça marche. = It works / Okay / That’s fine.

So in this sentence, marcher does not mean physical walking.

Both can mean it works, but they are not exactly the same in tone.

• ça marche is very common, natural, and slightly informal
• ça fonctionne is a bit more neutral or technical

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:

• an argument
• an explanation
• a theory
• a solution
• a plan

So the sentence does not mean that the proof itself is literally working like a machine. It means something like:

• With this proof, the argument works
• With this proof, it holds up
• With this proof, it makes sense

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:

• Avec cette preuve, ça marche.
• Avec cette preuve ça marche.

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.

Yes. That is grammatically correct.

• ça marche = more everyday, more natural in speech
• cela marche = more formal, more written, sometimes a little stiffer

So both are possible:

• Avec cette preuve, ça marche.
• Avec cette preuve, cela marche.

In normal conversation, most speakers would prefer ça.

Yes. Preuve can mean proof or evidence, depending on context.

That is why this sentence could suggest:

• a formal proof
• supporting evidence
• something that confirms a claim

The exact English word depends on what kind of situation is being discussed.

Sometimes, but the nuance changes.

• avec cette preuve = neutral; with this proof / using this proof
• grâce à cette preuve = thanks to this proof; more clearly shows that the proof is the reason something succeeds

So:

• Avec cette preuve, ça marche. = neutral and flexible
• Grâce à cette preuve, ça marche. = more like thanks to this proof, it works

If you do not want to emphasize gratitude or a strongly positive cause, avec is safer.

It is fairly natural and everyday in tone, mainly because of ça marche.

The structure itself is normal French, but ça gives it a spoken, informal feel.

A more formal version might be:

• Avec cette preuve, cela fonctionne.
• Avec cette preuve, l’argument tient.

So the original sentence is correct, but it sounds more conversational than highly formal.