Reading the book Analysis I (Fourth Edition), by Terence Tao, I found, on page 78, Lemma 4.1.3 (Addition and multiplication are well-defined.)
The Lemma says:
Let a, b, a', b', c, d natural numbers. If (a — b) = (a' — b'), then (a — b) + (c — d) = (a' — b') + (c — d) and (a — b) x (c — d) = (a' — b') x (c — d), and also (c — d) + (a — b) = (c — d) + (a' — b') and (c — d ) x (a — b) = (c — d) x (a' — b'). Thus addition and multiplication are well-defined operations (equal inputs give equal outputs).
Next, the author presents a Proof.
Well... my question is whether the Lemma should also include, in the hypothesis, (c — d) = (c' — d'), with c' and d' being natural numbers.
Note: The definition of an integer is given, in the mentioned book, in 4.1.1:
An integer is an expression of the form a — b, where a and b are natural numbers. Two integers are considered to be equal, a — b = c — d, if and only if a + d = c + b.