An couple elementary examples of this:

Greatest Common Divisor.

Consider the following two statements:

S1: The greatest common divisor $d=\gcd(a,b)$ satisfies: (i) $d \mid a$ and $d\mid b$ and (ii) if $c \mid a$ and $c \mid b$ then $c \mid d$.

S2: The greatest common divisor $d=\gcd(a,b)$ is the least positive element of the set $\{ax+by | x,y \in \mathbb{Z}\}$.

I have seen textbooks define gcd with S1 and then prove S2 and I have seen textbooks define gcd with S2 and then prove S1.

Induction vs. Well-Ordering Principles

Likewise, axiomatic constructions of the natural numbers can choose to take the Principle of Induction: $$((a \in S) \land ((\forall n \ge a)(n \in S \Rightarrow n+1 \in S))\Rightarrow \{a,a+1,a+2,a+3,...\} \subseteq S$$ or the Well-Ordering Principle: $$(\forall S \in \mathcal{P}(\mathbb{N}), S \neq \emptyset)(\exists a \in S)(\forall s \in S)(a \le s)$$ as an axiom and then prove the other as a theorem.

A couple elementary examples of this:

Greatest Common Divisor.

Consider the following two statements:

S1: The greatest common divisor $d=\gcd(a,b)$ satisfies: (i) $d \mid a$ and $d\mid b$ and (ii) if $c \mid a$ and $c \mid b$ then $c \mid d$.

S2: The greatest common divisor $d=\gcd(a,b)$ is the least positive element of the set $\{ax+by \mathrel\vert x,y \in \mathbb{Z}\}$.

I have seen textbooks define gcd with S1 and then prove S2 and I have seen textbooks define gcd with S2 and then prove S1.

Induction vs. Well-Ordering Principles

Likewise, axiomatic constructions of the natural numbers can choose to take the Principle of Induction: $$((a \in S) \land ((\forall n \ge a)(n \in S \Rightarrow n+1 \in S))\Rightarrow \{a,a+1,a+2,a+3,\dotsc\} \subseteq S$$ or the Well-Ordering Principle: $$(\forall S \in \mathcal{P}(\mathbb{N}), S \neq \emptyset)(\exists a \in S)(\forall s \in S)(a \le s)$$ as an axiom and then prove the other as a theorem.