Proof of compliance for the relative operator on the proportional distribution of unused share in an ordering fairshare system

Decentralized prioritization is a technique to in grid fairshare scheduling without centralized control. The technique uses an algorithm that measures individual user distances between quota allocations and historical resource usage (intended and current system state) to establish a semantic for prioritization. When a a user in a subgroup doesn't use its share, it is desired that its corresponding usage time is divided among the active users proportionally to their corresponding target usage. This report is the mathematical proof for the relative operator on the proportional distribution of unused share among the users of the same subgroup