Corresponding block bimodules are Green correspondents

If $(x,y)$ normalizes $\mathrm{\Delta }D$, conjugation and coordinate projection give $xD{x}^{-1}=yD{y}^{-1}=D$. Thus $x,y\in N$ and $N_{G\times G}(\mathrm{\Delta }D)\le N\times N$. By F1, F2 and F4 both blocks are nonzero indecomposable modules with vertex exactly $\mathrm{\Delta }D$, and the defining induction property gives $b\mid {\mathrm{Res}}_{N\times N}B$. The global vertex is already proved by the Brauer bijection; it is not inferred from this summand relation.

Apply F3 under A1 using the normalizer containment in step 1.1. Its unique vertex-$\mathrm{\Delta }D$ restriction summand must be $b$, and the inverse correspondence sends $b$ to $B$. These are exactly the two Green-correspondence assertions. When $D=1$ or $N=G$, the double-group normalizer interval is the identity case and the correspondence fixes the block. The only added AC use is application of F3, whose inherited finite-length argument declares it; the finite group calculation in step 1.1 and the block bijection do not use AC.