  We lay down an elementary yet fundamental lemma concerning a finite
algebraicness property of a smooth map from an Azumaya/matrix manifold with a
fundamental module to a smooth manifold. This gives us a starting point to
build a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometry
and calibrated geometry that are both tailored to and guided by D-brane
phenomena in string theory and along the line of our previous works D(11.1)
(arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).
