At any single flight condition for trimmed (hands-off) level flight there will one power setting and elevator trim position that is correct for the gross weight and desired airspeed.
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The only variables the pilot can alter are power and elevator position. (we will NOT consider special cases such as dumping fuel, etc.)
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Change the power setting, and the airplane is going to change something in the flight path because the energy in the system has changed.
Adding power.. the elevator trim position is now incorrect (not enough down).. and the airplane will accelerate upwards and forward, until it achieves a new attitude and/or altitude for which the new power and unchanged trim position are correct... if that is possible within the constraints of the airplane performance.
Decrease power, the airplane is also now at an untrimmed configuration, and will descend/pitch down (due to the unchanged elevator position) again seeking a trimmed flight condition, which might not also be possible..
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Power controls altitude.
Elevator controls airspeed.
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"But, but... I pull up and the plane goes up..."
Yes. And it slows down as it does so. That's why looping from level flight can result in a stall at the top of the loop.
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(Except for TMs, which can't get to the top of a loop, and stall going vertically or fall off to the side.

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Unless energy is -added- to the system by increasing power, the change in airspeed can result in loss of flying speed.
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Downthrust is a method of counteracting some undesired characteristic of the plane.. generally on a high-wing, it counters the drag of the wing as speeds increases. Up to a point. No single fixed control can be effective at all flight speeds.
On a Kadet for instance with its 4 degrees of downthrust, flying at full (over)power in level flight the elevator will be down about 10 degrees to keep the plane level. Suddenly reduce the power.. less thrust, the wing is still lifting/dragging because the speed hasn't had time to change, the plane's nose goes up. Until it slows down to an airspeed suitable for the new thrust level.