Proof of Lyapunov Matrix Equation I read the book A Linear Systems Primer by Antsaklis, Panos J., and Anthony N. Michel (Vol. 1. Boston: Birkhäuser, 2007) and I am confused by a part of the proof of a theorem about the Lyapunov Matrix Equation.

\begin{equation} 
\dot{x}=Ax\tag{4.22} 
\end{equation}
  \begin{equation} 
C=A^TP+PA\tag{4.25} 
\end{equation}
  Theorem 4.22. The equilibrium $x = 0$ of (4.22) is stable if there exists a real,symmetric, and positive definite $n \times n$ matrix $P$ such that the matrix $C$ given in (4.25) is negative semidefinite. 
Proof. Along any solution $ \phi(t,x_0) \triangleq \phi(t)$ of (4.22) with $\phi(0,x_0)=\phi(0)=x_0$, we have
\begin{equation} 
\phi(t)^TP\phi(t)=x_0^TPx_0+\int_0^t \frac{d}{d\eta}\phi(\eta)^TP\phi(\eta)d\eta\\
=x_0^TPx_0+\int_0^t \phi(\eta)^TC\phi(\eta)d\eta \tag{*}
\end{equation}
for all $t \ge 0$........

So how to obtain (*)? 

This is a variant of the fundamental theorem of calculus i.e. for an absolutely continuous function $f$ on $[0,T]$ we have for $t \in [0,T]$ 
$$f(t)=f(0)+\int_0^t \frac{d}{dy} f(y) ~dy$$
So in your case for $f(t)=\phi(t)^T P\phi(t)$ we get
\begin{align} \phi(t)^T P\phi(t)&=\phi(0)^T P\phi(0)+\int_0^t \frac{d}{dy} \phi(y)^T P\phi(y) ~dy \\ &=x_0^T Px_0+\int_0^t \frac{d}{dy} \phi(y)^T P\phi(y) ~dy \end{align}
