How would you prove the first law of thermodynamics?

Because EVERY known chemical and physical process is known to follow that law, why can't it be proven? There are various formulations in which there is a sign convention relating to work done on or by the system.

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system; it can only change forms. This law can be proven experimentally by conducting various experiments where energy transformations occur and measuring the energy input and output. Additionally, theoretical derivations based on the principles of conservation of energy and the laws of thermodynamics can provide further evidence for the first law. Overall, experimental observations and theoretical analysis support the validity of the first law of thermodynamics.