Expression for Integers as Powers of Same Primes

Theorem
Let $a, b \in \Z$ be integers.

Let their prime decompositions be given by:

Then there exist prime numbers:
 * $t_1 < t_2 < \dotsb < t_v$

such that:

Proof
In the prime decompositions $(1)$ and $(2)$, we have that:


 * $q_1 < q_2 < \dotsb < q_r$

and:
 * $s_1 < s_2 < \dotsb < s_u$

Hence we can define:

as all the $q_1, q_2, \dotsc, q_r$ are distinct, and all the $s_1, s_2, \dotsc, s_u$ are distinct.

Then let:
 * $T = E \cup F$

and let the elements of $T$ be renamed as:
 * $T = \set {t_1, t_2, \ldots, t_v}$

where all the $t_1, t_2, \dotsc, t_v$ are distinct, and:
 * $t_1 < t_2 < \dotsb < t_v$

Let $\iota: E \to T$ be the inclusion mapping:
 * $\forall q_i \in E: \map \iota {q_i} = q_i$

Let $\iota: F \to T$ be the inclusion mapping:
 * $\forall s_i \in F: \map \iota {s_i} = s_i$

Then we have that:

and:

Thus $a$ and $b$ can be expressed as the product of powers of the same primes, on the understanding that one or more of the powers in either product may be zero.