The usual "proof" entropy is a state property is like that:
"Consider a system which undergoes a reversible process from state 1 to state 2 along path A, and let cycle be completed along path B, which is also reversible. Since the cycle is reversible we can write:
$$\int_1^2 \delta Q / T + \int_2^1 \delta Q / T = 0 $$
Now let cycle be completed along path C, but paths B and C represent arbitrary reversible processes. So $\int_2^1 \delta Q / T $ is the same for all reversible paths between states 2 and 1."
My question is, isn't the equation above already assume entropy is a state property? Only if it is a state property it can go around a cycle without changes. How can it be valid to prove entropy is state property if it is already assumed from the beginning?