Lemma 29 ([Index quotients]). The positive integers e_ ij =|P_ ij |/|T_i| satisfy e_ ik =e_ ij e_ jk for every i<j<k . They multiply in the same way through each selected shortcut.

The active factor T_i is a composition factor of P_ ij , so its order divides |P_ ij | . Saturation and supplementation give [z_j:z_i]=[T_j^m:P_ ij ^m] =(|T_j| |P_ ij | )^m. Multiplying group indices through z_j and taking the positive m th root gives |T_k| |P_ ik | =|T_k| |P_ jk | |T_j| |P_ ij | . Rearranging and dividing by |T_i| proves the formula. The same calculation applies at a shortcut because its three multiplicities agree. No identification of its intermediate label with a chain label is required.