I read the book A Linear Systems Primer by Antsaklis, Panos J., and Anthony N. Michel (Vol. 1. Boston: Birkhäuser, 2007) and I think a part of a theorem about the Lyapunov Matrix Equation seems wordy.
\begin{equation} \dot{x}=Ax\tag{4.22} \end{equation} Theorem 4.29. Assume that the matrix A [for system (4.22)] has no eigenvalues with real part equal to zero. If all the eigenvalues of A have negative real parts, or if at least one of the eigenvalues of A has a positive real part,then there exists a quadratic Lyapunov function \begin{equation} v(x)=x^TPx,P=P^T \end{equation} whose derivative along the solutions of (4.22) is definite (i.e., it is either negative definite or positive definite).
At the beginning of the theorem, it says "A has no eigenvalues with real part equal to zero" which means the eigenvalues of A either have negative real part or positive real part. So I think the sentence following this (marked bold in the theorem) is redundant and makes the theorem reads wordy. Am I right? Is it really necessary to write the bold sentence in this theorem?