*. We-assume-that-h and . Nori-semistable, Then s * i ? * h * V = u * i * V is Nori-semistable. We will deduce from this that i * V is Nori-semistable. So one is reduced to prove the statement for a finite faithfully flat morphism h : Z ?? Y , where Z and Y are smooth projective curves

. Proof, According to Lemma 6.2, V is Nori-semistable if and only if h * V is Nori-semistable, and according to Corollary 4.4 this last condition is equivalent to the fact that g * h * V is Nori-semistable. The last statement is an immediate consequence of Theorem 3

. Proof, If V is essentially finite, h * V is also essentially finite, and according to Theorem 5

*. and . Essentially-finite, Conversely if g * h * V is essentially finite, h * V is essentially finite according to Theorem 5

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