The phrase "by the relevant documents" suggests that the documents served as evidence to support the insurance claim.
The phrasal verb "backed up" means to support with evidence or proof, making it the most suitable choice in this context.
Option (A) "sustained" means to maintain or uphold but is less natural when specifically referring to documents as evidence.
Option (B) "backed out" means to withdraw from a commitment, which is the opposite meaning.
Option (D) "proved" is acceptable in meaning but lacks the idiomatic quality that "backed up" provides for formal English usage in such cases.
Thus, "backed up" is the best option for expressing evidence-based support for the claim.