According to the fundamental theorem of arithmetic:
$$ \begin{align} a &= {p_1}^{x_1} {p_2}^{x_2} {p_3}^{x_3} \cdots {p_n}^{x_n} \\ b &= {p_1}^{y_1} {p_2}^{y_2} {p_3}^{y_3} \cdots {p_n}^{y_n} \\ (a,b) &= {p_1}^{\min(x_1,y_1)} {p_2}^{\min(x_2,y_2)} {p_3}^{\min(x_3,y_3)} \cdots{p_n}^{\min(x_n,y_n)} \\ [a,b] &= {p_1}^{\max(x_1,y_1)} {p_2}^{\max(x_2,y_2)} {p_3}^{\max(x_3,y_3)} \cdots{p_n}^{\max(x_n,y_n)} \\ c &= {p_1}^{z_1} {p_2}^{z_2} {p_3}^{z_3} \cdots {p_n}^{z_n} \end{align} $$
One of the properties of the \(\min\) and \(\max\) is \(\max(\min(m,n), k) = \min(\max(m,k), \max(n, k))\). For example:
$$\begin{gather} \max(\min(1,3), 5) = \min(\max(1,5), \max(3, 5)) = 5 \\ \max(\min(8,9), 5) = \min(\max(8,5), \max(9, 5)) = 8 \\ \max(\min(4,6), 5) = \min(\max(4,5), \max(6, 5)) = 5 \end{gather}$$
By definition of \(\operatorname{lcm}\), we can write \([(a,b),c]\) as:
$$\begin{align} [(a,b), c] &= p_1^{\max(\min(x_1,y_1), z_1)} p_2^{\max(\min(x_2,y_2), z_2)} \cdots p_n^{\max(\min(x_n,y_n),z_n)} \\ &= p_1^{\min(\max(x_1,z_1), \max(y_1,z_1))} p_2^{\min(\max(x_2,z_2), \max(y_2,z_2))} \cdots p_n^{\min(\max(x_n,z_n), \max(y_n,z_n))} \end{align}$$
However:
$$ \begin{align} [a,c] &= p_1^{\max(x_1,z_1)} p_2^{\max(x_2,z_2)} p_3^{\max(x_3,z_3)} \cdots p_n^{\max(x_n,z_n)} \\ [b,c] &= p_1^{\max(y_1,z_1)} p_2^{\max(y_2,z_2)} p_3^{\max(y_3,z_3)} \cdots p_n^{\max(y_n,z_n)} \end{align} $$
By definition of \(\gcd\):
$$ ([a,c], [b,c]) = p_1^{\min(\max(x_1,z_1), \max(y_1,z_1))} p_2^{\min(\max(x_2,z_2), \max(y_2,z_2))} \cdots p_n^{\min(\max(x_n,z_n), \max(y_n,z_n))} $$
This shows that \([(a, b), c] = ([a, c], [b, c])\). A similar proof can be made to show that \(([a, b], c) = [(a, c), (b, c)]\).