Lemma: Anchored sum lemma

For subspaces $B,P\subset A$ with $B\cap P=0$,

Proof. Define $\pi (p+b)=p$ on $P\oplus B$, and extend it algebraically to $A$. Then $\pi \otimes \pi$ is identity on $P\otimes P$ and zero on the anchored sum. An element in the intersection is both preserved and annihilated, hence zero. No continuity of $\pi$ is asserted.