Result
R3749: Partition of real square matrix into sum of symmetric and antisymmetric parts allows us to write
\begin{equation}
A = \frac{A + A^T}{2} + \frac{A - A^T}{2}
\end{equation}
Multiplying from the left by $x^T$ and from the right by $x$ and applying
R3745: Real square matrix antisymmetric part is zero definite, we then have
\begin{equation}
x^T A x
= x^T \left( \frac{A + A^T}{2} \right) x + x^T \left( \frac{A - A^T}{2} \right) x
= x^T \left( \frac{A + A^T}{2} \right) x
\end{equation}
$\square$
