What is the relation between trim setting and constant airspeed?
Your friend is right.  Elevator trim isn't setting a pitch, it is compensating for control forces on the elevator, which vary with speed.  If you are trimmed for level flight at 100 kias and reduce power the plane will initially slow.  As it slows the trim will cause a nose-down moment causing the plane to pitch down.  Given no input, the plane will equilibrate at whatever nose-down pitch is needed to regain the forces it was trimmed for, which means speeding back up to 100 kias. 

You are correct that to maintain 100 kias with reduced power the nose should be lowered, but the trim will take care of that for you.  

You might have a misconception about how trim works, based on your argument.  A typical trim setup in light airplanes is essentially another control surface on the elevator:



The elevator is the large control surface and that smaller control surface is the elevator trim.  When you adjust the pitch trim you are adjusting this trim tab which provides a force on the elevator that counters aerodynamic forces.  This is why the airplane responds in pitch to maintain constant aerodynamic forces (e.g. constant indicated airspeed) for a given trim setting.



You can test this in the airplane easily.  Take your plane up to the altitude you practice maneuvers at, set 75% power (or some other number) and trim for straight and level.  Add power and watch the plane climb, decrease power and watch it descent, both without touching the trim.  Go back to straight and level and trimmed and accelerate without climbing -- note you are having to retrim as your speed increases.  Decelerate without descending  -- note you are retrimming as your speed decreases.