Lemma 1 (Zassenhaus)

Let H and K be two stable subgroups of a group with operators G and H' and K' normal stable subgroups of H and K respectively; then H'.(H \cap K') is a normal stable subgroup of H'.(H \cap K), K'.(K \cap H') is a normal stable subgroup of K'.(K \cap H) and the quotient groups with operators (H'.(H \cap K))/(H'.(H \cap K')) and (K'.(K \cap H))/(K'.(K \cap H')) are isomorphic.

By Theorem 4, H' \cap K = H' \cap (H \cap K) is a normal stable subgroup of H \cap K; similarly K' \cap H is a normal stable subgroup of K \cap H; hence (no. 6, Corollary 2) (H' \cap K)(K' \cap H) is a normal stable subgroup of H \cap K. By Theorem 4 applied to the group H,

is a normal stable subgroup of H'.(H \cap K) and the quotient group

is isomorphic to

In the last quotient, H and H' on the one hand and K and K' on the other appear symmetrically; permuting them the stated result is obtained.