Inverse element is not proof, but definition. Define the inverse element, and then define the group. The algebraic structure of a group must have inverse elements.

The reason why inverse element is defined is because a kind of quite extensive and meaningful algebraic system can be abstracted into a system with unified inverse element properties.

This is like giving a definition to people, saying that people are a kind of creature, with the attributes of walking upright, using tools and knowing language. These properties, as well as the unit element and inverse element of algebraic structure, are used to describe classification and cannot be proved.