No, the final horizontal velocity is not always equal to the horizontal initial velocity. It depends on the circumstances and forces acting on the object.
In the absence of any external forces that would change the horizontal velocity, such as air resistance or a force acting on the object, the final horizontal velocity would indeed be equal to the horizontal initial velocity.
However, if there are external forces acting on the object, such as friction or a force pushing or pulling the object horizontally, the final horizontal velocity can be different from the initial velocity. These external forces can accelerate or decelerate the object, resulting in a change in its horizontal velocity.
For example, if a car is subject to frictional forces and its brakes are applied, the car's initial horizontal velocity will decrease, and the final horizontal velocity will be lower than the initial velocity. On the other hand, if a force is applied to accelerate the car, the final horizontal velocity will be greater than the initial velocity.
In summary, the final horizontal velocity can be equal to the initial velocity if no external forces act on the object, but it can change if external forces are present.