I find the following paragraph at the end of pag.4 of the PBFT paper by Castro and Likov confusing:

This is true because $prepared(m,v,n,i)$ and $|\mathcal{R}|=3f+1$ imply that at least $f+1$ non-faulty replicas have sent a pre-prepare or prepare for $m$ in view $v$ with sequence number $n$.

Based on my understanding, the clause "and $|\mathcal{R}|=3f+1$" is not required for the implication to be true.
$prepared(m,v,n,i)$ indicates that the replica $i$ has received $2f$ prepares from different backup that match the accepted pre-prepare for message $m$, in view $v$ with sequence number $n$. Out of the $2f+1$ pre-prepare and prepare messages received, at most $f$ messages are from faulty replicas, which leaves at least $f+1$ messages from non-faulty replicas. So, I really do not understand why the paper also includes the clause "and $|\mathcal{R}|=3f+1$".


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