Proposition: Conditional interval snapshot comparison

In the preceding specialization, if

then the returned finite state depends only on the normal input state's restriction to $A_{\lambda }(I)$. Conversely, equality of returned states for every pair of normal equal-snapshot inputs implies (eq:netbridge).

Proof. Apply the normal version of Theorem thm:comparison with $M_{\mathrm{in}}=\mathcal{B}(H_{\lambda })$, $A_{\mathrm{in}}=A_{\lambda }(I)$, and $M_{\mathrm{out}}=A_{\mathrm{out}}=\mathcal{B}({\mathbb{C}}^2)$.