My books said that energy is a quantity that:
So, can (2) be inferred from (1)? And how to proof (2) from (1) or from something else ?
The words mechanical and internal energy here are pretty ambiguous and are meant to just broadly separate the energy of a system into degrees of freedom of the system as a whole (mechanical) versus degrees of freedom of its constituent atoms/molecules (internal), and I think that's all your book is trying to get you to think about.
For example, we would say that a ball at rest on the ground has no mechanical energy because it has no velocity or gravitational potential. But, of course this is not the full story because the ball has internal energy because it is actually made up of atoms/molecules which are of course not perfectly still.
Then, statement (2) is trivial because if the system has some energy it must be "internal" or "mechanical", there is simply no other option.
(1) is a statement of what's called the work-energy theorem, and I think the top answer on this thread does a nice job of explaining why it's emphasized in introductory mechanics:
The energy of a system includes other components, for example the electromagnetic energy.
How to prove it? You do experiments, Joule's Experiment, for example. Physics is not mathematics: in physics we prove things by testing them in experiments. We have no axioms to base a mathematical proof on.
Without seeing the full context of the bulleted statements in your book, it appears bullet 1 is a special case of bullet 2.
Bullet 1:
The first bullet appears to be a statement of the conservation of mechanical energy, which states the variation in the sum of the kinetic and potential energy of a system equals the net external (to the system) work done on the system, assuming no dissipative forces (e.g., friction) acting within the system. Or
$$\Delta KE+\Delta PE=W$$
For an isolated system, $W=0$ and $\Delta KE+\Delta PE=0$.
One reason why the statement is restricted to mechanical energy is it doesn't take into account the other means of causing a variation in energy which is covered in thermodynamics, namely heat, with is energy transfer due solely to temperature difference.
The other reason is it only applies to the kinetic and potential energies of objects at the macroscopic level, i.e, the visible motion and position of objects with respect to a particular frame of reference. An example is a ball having kinetic energy and gravitational potential energy due to its velocity and position with respect to the ground. It does not cover kinetic energy of the motion of the molecules of the ball (reflected by its temperature) or the molecular potential energy due to the intermolecular forces of the molecules of the ball. Those energy forms are the internal energy of the ball.
Bullet 2:
The second bullet is a statement of conservation of energy of all forms, mechanical energy plus internal energy, not just mechanical energy. By including internal energy it includes energy at the atomic and molecular level. For example, when friction occurs in a mechanical system, mechanical energy is lost due to conversion into internal energy reflected by an increase in temperature. So while mechanical energy is not conserved, total energy of the system is conserved. Consequently, bullet one can be considered a special case of bullet 2.
Hope this helps.
