Corollary 3

Let M, N, and P be semiprimordial modules. Suppose that $M\oplus P$ is isomorphic to $N\oplus P$ and that [P : L] is finite for every primordial module L. Then M and N are isomorphic.

By assumption, we have

[M : L] + [P : L] = [N : L] + [P : L]

for every primordial module L. Since [P : L] is finite, it follows by induction from (Set Theory, III, §3, No. 4, p. 162, Proposition 8) that we have [M : L] = [N : L] for every primordial module L. The modules M and N are therefore isomorphic by Corollary 1.