The proof is straightforward. According to Newton's second law of motion, the resultant force must produce the change in the momentum of the fluid that occupies the portion of the filament between and at any given instant of time, . Suppose that at time the fluid in question occupies the portion of the filament between and . The momentum of the fluid in question has then increased by the momentum of the fluid between and , and decreased by the momentum of the fluid between and . Hence, there has been a gain of momentum at , and a loss of momentum at . Thus, the net rate of charge of momentum consists of a gain at , and a loss at . This net rate of change is produced solely by the thrusts acting on the walls and ends of the filaments. It follows that these thrusts are equivalent to the forces and acting normally outward at and , respectively.
If and are the pressures at and , respectively, then the thrusts acting normally inward on the ends of the filament are at and at . According to Euler's theorem, the thrusts exerted on the walls plus the thrusts acting on the ends are equivalent to the normal outward forces at and at . It follows that the thrusts exerted by the walls on the fluid are equivalent to the normal outward forces at and at . Conversely, the thrusts exerted by the fluid on the walls are equivalent to normal inward forces at and at .
Note, finally, that the Euler momentum theorem obviously also applies to a stream tube, as long as the flow through the ends of the tube is uniform across the cross-section.