After $(11)$, $e^{-s}$ disappear. To me, you can only remove that term and the inequality still stays valid, if after removing, it makes the LHS smaller. However, it is not obvious to me that such guarantee exists for this proof.
Am I missing anything? Or is this proof wrong?
This proof is from the paper https://arxiv.org/abs/1801.09414.
Plugging (13) into (12) yields that $$\frac{1}{C(C-1)}\sum_{i,j,i\neq j}e^{s(W_i^TW_j)}\geq e^{-\frac{s}{C-1}}.$$ Plugging this into (11) yields that $$\frac{1}{P_W}\geq 1+\frac{e^{-s}}{C}\sum_{i,j,i\neq j}e^{s(W_i^TW_j)} \geq 1+(C-1)e^{-s}e^{-\frac{s}{C-1}}=1+(C-1)e^{-\frac{sC}{C-1}},$$ which is precisely what (14) says.