I've been reading Milnor's notes on algebraic K-theory, and have trouble understanding the last step of the proof of Lemma 3.2. Here is the set up.
Let $A$ be a ring, and $P$ a finitely generated protective $A$-module. Define a map $$\operatorname{Aut}(P) \to K_1 A$$ as follows. Choose $Q$ so that $P\oplus Q$ is free and send $\phi$ to $\phi\oplus 1_Q$, which lives in $GL(A^n)$ for some $n$.
The lemma is just to show this is well defined, so showing that a different choice of basis for $P\oplus Q$ or a different choice if $Q$ just conjugates in $GL(A^n)$.
He writes that if we choose $Q'$ instead, with $P\oplus Q' \cong A^t$, then $Q \oplus A^t \cong Q' \oplus A^n$, which I understand. But then he just says that it follows that changing $Q$ only alters the embedding by an inner automorphism. How does this follow?