How does the engine produce aerodynamic lift at high angle of attack? This answer asserts that "at high angle of attack the engine is producing aerodynamic lift". In the context of that answer, we are concerned about the increased torque that raises the nose of the aeroplane with respect to the center of gravity, so we should narrow down our interpretation of the otherwise vague word "lift" as the force component that is perpendicular to the wing or fuselage of an aeroplane. We define the angle of attack as the angle between the fuselage and the incoming airflow direction at infinity (shorthand word for "reasonably far away"). It is very different from the angle of the fuselage makes with the horizontal at high speed while very close to the latter at very low speed. It puzzles me in two ways. 


It claims the engine itself produces fuselage-perpendicular force in addition to the thrust parallel to the fuselage and wing, despite the thrust of the engine is directed parallel to the fuselage and thus the wing. How is that perpendicular force produced?
It claims that fuselage-perpendicular force thus produced is dependent on the angle of attack. How does that work? 


I would like to see a technical elucidation of this claim or some references supporting or refuting it, best if they come with mathematical derivation.



Edit: 

Some answers below observe that when the plane moves, without engine turned on, against the airflow, part of the airflow curves downwards to conclude that a net force perpendicular to the fuselage is exerted. It is not that simple as looking at part of the flow stream, since part of the flow stream goes upwards above the separating point. The net force is the integration of all the pressure on the whole surface, and the pressure on the opposite side of the object presses in the opposite direction. In fact, in a potential flow, the D'Alembert's paradox  shows that the net pressure is exactly zero. Of course, real airflow is not potential flow, but this refutes the overly simple rationale of using downward airflow to derive the fuselage-perpendicular force.

However, when the engine is turned on, especially at high power. The situation may be different so long as the airflow is consistently flowing in the downward direction long after exiting the engine. The momentum of the airflow through the engine may overwhelm other parts of the airflow. But the argument must be more sophisticated than just the downward direction of the airflow.
Anything, even a metal plate creates "lift" when it is under an angle with an air flow. Lift isn't a magical attribute to "wings", it is just a physical consequence of the law of conservation of momentum: when you push a stream of air down, the reactive force is upwards.

the fuselage/engines also act as minor lifting surfaces: when they are under an angle they, inefficiently, redirect the airflow downwards. 

For the engine this is also a bit due to the definition of thrust: thrust equations consider an inlet in the same direction of the thrust vector. So the inlet &amp; compressor redirect the flow "downwards" under high angles of attack. - This effect has to show up somewhere: as a lift force generated by the engine.