Let's define \(d_1\) and \(d_2\):
$$ (a_1, a_2, a_3 \ldots, a_n) = d_1 $$$$ ((a_1, a_2), a_3, \ldots, a_n) = d_2 $$
If \(d_1\) divides \(a_1\) and \(a_2\), then it also divides \((a_1, a_2)\):
\[\displaylines{ d_1 | (a_1, a_2) \\ d_1|a_3, \ d_1|a_4, \ \ldots , \ d_1|a_n \\ \therefore d_1|d_2} \]
If \(d_2\) divides \((a_1, a_2)\), then:
\[\displaylines{d_2|a_1 \text{ and } d_2|a_2 \\ d_2|a_3, \ \ldots , \ d_2|a_n \\ \therefore d_2 | d_1 } \]
If \(d_1 | d_2\) and \(d_2 | d_1\), then \(d_1 = d_2\).