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Q2: active manifold M is dependent on x [...] A: The manifold M in Theorem 3.1 is the manifold associated to x^*, to clarify this, we will denote it as M_{x^*}. Q3: why applying the Baillon-Haddad theorem [...] A: The subdifferential partial Phi is a set-valued operator and *cannot* be non-expansive in the classical sense. It is indeed non-expansiveness of G_k that we need, and this is a consequence of Baillon-Haddad theorem which states that \nabla F is firmly non-expansive, and thus that G_k is \alpha_k-averaged, hence non-expansive, for the prescribed range of \gamma_k.