Physics is an observational science. It measures and observes the way nature behaves, and ever since Newton, models these measurements and observations with mathematics.
Mathematics is a discipline where one starts with axioms uses logic and arrives to theorems and expressions that can be proven using the axioms and other theorems. One ends a proof in mathematics with QED ( quod erat demostrandum, i.e. which should be demonstrated). Proofs belong to mathematics.
Physics in addition to the mathematical axioms that control the mathematical formulas used, has postulates, or laws, that make a one to one correspondence with physical measurements and observations. The postulates/laws are a distillate of innumerable observations and measurements. They can never be proven, they are assumed. In contrast to a mathematical system, the models of physics are validated, i.e. follow the mathematical predictions usually for certain regions of variables. Newtonian laws are not useful for high energies where special relativity or general relatitivy postulates had to be devised to explain observations.
A physics theory can be falsified , but not proven.
Your question depends on the validity of Newton's laws of motion, and the Galilean transformations. These work exceptionally well in everyday life, so there they are validated and all consequences of the mathematical models are expected to hold. They are falsified for elementary particles and high energies and for high gravitational masses where other postulates were posited so that observations could be fitted with a mathematical model.