Conditional typical-set fan cardinality bound

By [F4], choose $\delta >0$ so that every joint probability vector $q$ with $\mathrm{\parallel }q-P{\mathrm{\parallel }}_{\mathrm{\infty }}\le \delta$ satisfies $\mathrm{\mid }{H}_q(X_S\mid X_C)-H_P(X_S\mid X_C)\mathrm{\mid }\le \eta$. The joint alphabet is identified with ${\mathcal{X}}_S\times {\mathcal{X}}_C$ by a coordinate permutation, so the same coordinatewise neighborhood applies to the joint type of $(x_S^n,x_C^n)$.

For the fixed $x_C^n$, each $x_S^n$ in the set has a unique joint empirical type $q$ by [F2]. There are at most $(n+1{)}^m$ possible such types, since each of the $m$ joint symbol counts is an integer from $0$ to $n$; the sum constraint can only reduce this count.

For each feasible type $q$ arising from a typical block, [F1] and step 1.1 give $H_q(X_S\mid X_C)\le H_P(X_S\mid X_C)+\eta$. By [F3], its conditional type class has size at most $\exp (n(H_P(X_S\mid X_C)+\eta ))$. Summing this bound over the at most $(n+1{)}^m$ types from step 1.2 proves the claimed inequality. The empty-complement case uses its singleton alphabet and the same count.