A paid for B's patented product B 1 and installed it on its own patented product A. Is there any reason for B to claim patent A?

According to your description, A bought B 1, installed it on product A and sold it. Whether B has reason to claim the right of A patent depends on the following factors:

Scope of patent right: firstly, it is necessary to determine whether the description in the patent claim of product A of A defines a specific class B or whether it can be interpreted as including B 1. If it is clearly stipulated in the patent claim that there is only a specific category B, then B may have reason to claim the right to patent A, because A's products use B 1 which is not within the scope of patent right.

Novelty and non-dominant features: Party B shall prove that Party A's product A uses Class B technology owned by Party B instead of Party A's B 1 ... This may require patent comparison and technical analysis to determine whether B 1 covers the same novel and non-explicit features as Class B owned by B. ..

Infringement claim: Party B also needs to prove that Party A's product A uses Class B technology owned by Party B, and this use has caused economic losses to Party B ... This may require providing evidence and professional advice to prove the existence of patent infringement.

To sum up, Party B may have reason to claim the right of patent A, but the specific situation requires a detailed analysis and evaluation of the patent claim, Party A's products and Party B's patents ... It is recommended to consult a professional intellectual property lawyer for more specific and reliable legal advice.