Graduate Geometry Topology and Physics: Nakahara, Chapt. 1: Weyl Ordering

[XiYi]

In the text, the author try to deduce proposition 1.2, here is the detail (all in one dimention):

My question is the last line of formula, how to deduce it from the previous line? Here is a note of the author:

I understand the similar part of Srednicki QFT at chap 6 and I could get the point of nakahara, but can't seize the logic. It bother me a lot. Thanks for giving help.

 

[SergejMaterov]

Trotter splits $e^{−i(T+V)ε}$ into “purely kinetic” and “purely potential”.
The Gaussian integral yields the leading term $exp[im(x−y)^2/(2ε)]$ and normalization.
Taylor potential around the midpoint 𝑧=(𝑥+𝑦)/2 yields −iεV((x+y)/2) plus $𝑂(𝜀(𝑥−𝑦)^2)$.
All remaining inconsistencies are $𝑂(ε^2)$.
Weyl ordering — to emphasize that the Hamiltonian is arranged symmetrically, and when moving to the end of the integral over phase space, exactly the average value of the potential 𝑉 appears on the segment between 𝑥 and 𝑦.