Yes, you can use 'prove' or 'proof' with claim. It would be the same as saying 'Show me the evidence to back up what you're saying' example: You need to provide proof of your claim. Also: You need to show evidence to prove your claim. Or simply: You need to prove your claim.

Yes you can say"prove your claim" but you cannot say"proof your claim" To prove = show proof of, show evidence. To proof as two meanings. 1.to make something first before then going onto to make more of it. Often used in printing, where the printer(s) ask the client to verify "a proof" which they have to sign is correct. only then does the printer(s) print off the full amount. for book publishers they employ "proof readers" to read a book first before it is publicised "made public". 2. "The proof" = the evidence that establishes the fact(s). sentences: "prove your case" "have you proven your case" "the proof you have submitted proves your case" "you will prove your case only when you have given us proof of your innocence" "have you provided evidence to prove your claim" "have you the proof that proves your claim" "have you the evidence that proves your claim" The evidence is the proof that you must prove.