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Why is the avagadro's number independent of substance?

In the book of Irey, Thermodynamics, in vol1, IA, page 4, it is given that

Molecular (Atomics) Weight(M) : The radio of the mass of a molecule (atom) of a substance m^*, to one twelfth of the mass of an atom of the most common isotope of carbon C^{12}.

Mole: A gram mole is the amount of a substance whose mass is equal to its molecular weight in grams. The mass of the one gram mole equals M grams.

Avogadro's number: Since the mass of a mole is proportional to molecular weight, there are definite numberer of molecules in any molar unit, called N_A.

And in the remark, the author mentioned that $N_A$ is independent of molar units, and we know that it is even independent of substance.

However, with these definitions, I cannot understand why $N_A$ should be independent of substance.

Question:

With these definitions, why is the constant $N_A$ independent of substance that is used to measure $N_A$ ?

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