The core argument posits that $B$ represents the optimal set of transactions, characterized by having the highest fee rate while including certain transactions ($inc$) and excluding others ($exc$), without considering their interdependencies or topology. This optimality is underlined by the principle that any transaction $t$ not in $B$ but with a higher fee rate than the entirety of $B$ would necessitate its inclusion in $B$ to enhance $B$'s overall fee rate. Conversely, any transaction within $B$ that possesses a lower fee rate than $B$'s average could be removed to further optimize $B$. This bidirectional criterion ensures $B$'s uniqueness as the subset fulfilling these conditions.

Further exploration of this concept introduces the impossibility of having two distinct subsets, $B_1$ and $B_2$, both satisfying the aforementioned property without contradiction. If one were to assume such a scenario where $B_1$ is a proper subset of the larger $B_2$, the transactions exclusive to $B_2$ would inherently possess fee rates either equal to or less than those in $B_1$ to maintain $B_2$'s status as a viable subset. However, this creates a paradox as these same transactions must also exhibit fee rates greater than $B_2$ (and by extension, $B_1$) based on the initial property, leading to an inconsistency since it would imply they should also be included in $B_1$.

The crux of proving $pot = B$ lies in demonstrating that $pot$ adheres to the established property defining $B$. Given that $B$ comprises $inc$ along with a specific "prefix" of transactions from an unordered set ($und$) arranged by decreasing fee rate, and similarly $pot$ is constructed following the same logic, the equality holds true. This is particularly evident when considering the inclusion and exclusion of transactions based on their fee rates relative to $pot$; only those transactions with a fee rate higher than $pot$'s are considered, ensuring $pot$’s consistency with $B$'s definitional properties.