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The following corollary can be obtained immediately from Proposition 2.9.

Proof The proof can be obtained immediately from Theorem 10.

They can be obtained immediately from Gentzen's sequent-rules for implication.

end{aligned} (H) This formula can be obtained immediately from the following theorem.

Thus, the required result can be obtained immediately from Theorem 3.2.

Thus, the desired result (2.1) can be obtained immediately from (2.2).

According to (4), the conclusion of this theorem is obtained immediately from Theorem 3 and Theorem 4.

The following corollary is obtained immediately from Theorem 1. Corollary 2. Let φ = (φ 1,..., φ n ) be a holomorphic self-map of U n and α > 0.

Similar to the proof of Theorem 3, the conclusion of this theorem is obtained immediately from Lemma 1, Lemma 2, Lemma 3, and Lemma 5.

Consequently, (D lambda)equiv0), which means m_(b,lambda)=-frac{varphi_{2}(b,lambda)}{varphi_{1}(b,lambda )}=- frac{tilde{varphi}_{2}(b,lambda)}{tilde{varphi}_{1}(b,lambda )}=tilde{m}_(b,lambda). (2.18) The result is obtained immediately from the uniqueness theorem in [17] and this completes the proof.

In this case, the optimal alignment cost is obtained immediately from applying Equation 5 for this specific sub-instance, as formulated by term III in the equation.