The same axioms are to hold for the inequality of distances as hold for the inequality of numbers.
Heyting arithmetic, after all, has exactly the same axioms as Peano arithmetic, and is the logical closure of those axioms under intuitionistic, rather than classical, logic.
Although each support function satisfies these same axioms, the further issue of which among them provides an appropriate measure of inductive support is not settled by the axioms alone.
It's safe because the theory has the same axioms at each type, so it has the same theorems at each type, so whenever one has a proof of a formula Φ one also has a proof of Φ+.
As we can see, an inner product module obeys the same axioms as an ordinary inner product space, except that the inner product takes values in a more general structure than in the field of complex numbers.
Hilbert C ∗ -modules form a category in between Banach spaces and Hilbert spaces and obey the same axioms as a Hilbert space except that the inner product takes values in a general C ∗ -algebra rather than in the complex numbers ℂ.
The very same axioms, applied to a ring R instead of a field F yield modules.
These associated entities are declared with equivalence (via "equivalent class" or "same individual" axioms).
