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Abstract

We give a transparent proof that difference Picard-Vessiot theory is a part of the general difference Galois theory. We apply the proof to iterative $q$-difference Picard-Vessiot theory to show that Picard-Vessiot theory for iterative $q$-difference field extensions is in the scope of the general Galois theory of Heiderich. We also show that Picard-Vessiot theory is commutative in the sense that studying linear difference-differential equations, no matter how twisted the operators are, we cannot encounter quantification of the Galois groupoid.