Prove that energy is a property

Let us consider a cyclic process performed by the system

$$\int(δQ - δW) = \int_{I_A}^{2} (δQ - δW) + \int_{I_C}^{2} (δQ - δW) = 0 $$

in which the summation around the cyclic procedure has been divide into its component parts. Let us now consider a second cyclic process, that varies from the first in that the outward path is now B instead of A. Applying the First Law to this procedure, we acquire:

$$\int_{I_B}^{2} (δQ - δW) + \int_{I_C}^{2} (δQ - δW) = 0$$

From examining the above two equations we see that

$$\int_{I_A}^{2} (δQ - δW) = \int_{I_B}^{2} (δQ - δW) $$

This illustrates that the integral of (δ Q - δ W) from state 1 to state 2 is similar for paths A and B. As these have been randomly selected, we say that the integral has similar value for any path between 1 and 2.

Representing ∫ δ Q by Q, the total heat transfer during the procedure from 1 to 2, and ∫ δ W by W, the total work during the similar change of state, we can say that (Q – W) has similar value for any path between 1 and 2. From the definition of energy, equation

$$Q – W = E_2 – E_1$$

Therefore it is seen that $(E_2 − E_1)$ has similar value for any path among 1 and 2. This entails that the value of $(E_2 − E_1)$ based only on the end states. Which means that energy is a property of a system.