Considering the airflow through the prop as in a tube, the prop generates thrust by accelerating the air in the direction of flight. The only way for the stators to generate extra thrust is to further accelerate the air. How do they do this?
Excellently put question, Dick. The extra thrust comes from a two-step process:
1) Capture the vortex energy by turning the air back parallel to the direction of flight. In the process of doing this, the stator vane array has a larger exit area than its entrance area, so the air is slowed. The energy capture is reflected in the static pressure rise.
2) Re-accelerate the air through an annular nozzle, maintaining the axial flow direction.
Both steps are necessary!