Proof of E=mc^2:
The equation E=mc^2 is derived from Einstein's theory of relativity, specifically the theory of special relativity. It states that the energy (E) of an object is equal to its mass (m) multiplied by the square of the speed of light (c). Here is a proof of this equation:

1. Equivalence of Energy and Mass:
According to Einstein's theory of relativity, energy and mass are interchangeable. This is expressed by the famous equation E=mc^2, where c is the speed of light in a vacuum (approximately 3 x 10^8 meters per second). This equation implies that energy can be converted into mass and vice versa.

2. Mass-Energy Equivalence:
The principle of mass-energy equivalence states that the energy possessed by an object at rest is equal to its rest mass times the square of the speed of light. This means that even at rest, an object has energy associated with its mass.

3. Energy of a Moving Object:
When an object is moving, it possesses both kinetic energy (energy due to motion) and rest mass energy. The total energy of the object is the sum of these two energies.

4. Relativistic Energy:
According to special relativity, the total energy of a moving object is given by the equation E = γmc^2, where γ is the Lorentz factor. The Lorentz factor depends on the velocity of the object and is given by the equation γ = 1/√(1 - (v^2/c^2)), where v is the velocity of the object.

5. Rest Mass Energy and Kinetic Energy:
At rest, the velocity of an object is zero, so the Lorentz factor becomes γ = 1. In this case, the equation simplifies to E = mc^2, which represents the energy associated with the mass of the object at rest. This is the famous equation derived by Einstein.

6. Conclusion:
Therefore, the equation E=mc^2 shows the equivalence of energy and mass, and it holds true for both objects at rest and objects in motion. It is a fundamental equation in physics and has been experimentally verified numerous times.

Relation to the Mole Concept:
The mole concept is a fundamental concept in chemistry that relates the mass of a substance to the number of particles (atoms, molecules, or ions) it contains. It is based on Avogadro's constant, which states that one mole of any substance contains 6.022 x 10^23 particles.

The equation E=mc^2 is not directly related to the mole concept. The mole concept deals with the quantitative relationship between mass and the number of particles, while E=mc^2 deals with the equivalence of energy and mass. However, both concepts are important in understanding the behavior of matter and energy in different scientific contexts.