We can represent the prime factorization of \(a\), \(b\) and \(c\) like this:
$$ \begin{align} a &= {p_1}^{x_1} * {p_2}^{x_2} * {p_3}^{x_3}* \ldots * {p_n}^{x_n} \\ b &= {p_1}^{y_1} * {p_2}^{y_2} * {p_3}^{y_3} * \ldots * {p_n}^{y_n} \\ c &= {p_1}^{z_1} * {p_2}^{z_2} * {p_3}^{x_3}* \ldots *{p_n}^{z_n} \end{align} $$
If we take the \(\operatorname{lcm} \) of \(a\) and \(b\):
$$ \operatorname{lcm} (a, b) = {p_1}^{\max \{x_1, y_1\} } {p_2}^{\max \{x_2,y_2\} } {p_3}^{\max \{x_3,y_3\} } \ldots {p_n}^{\max \{x_n, y_n\} } $$
If we take the \(\operatorname{lcm} \) of \(\operatorname{lcm} (a, b)\) and \(c\):
$$ \operatorname{lcm} (\operatorname{lcm} (a, b), c) = {p_1}^{\max \{\max \{x_1, y_1\}, z_1\} } {p_2}^{\max \{ \max \{x_2, y_2\}, z_2\} } {p_3}^{\max \{ \max \{x_3, y_3\}, z_3\} } \ldots {p_n}^{\max \{ \max \{x_n, y_n\}, z_n\} } $$
It is trivial that:
$$ \max \{ \max \{ m, n \}, k \} = \max \{k, m, n\} $$
This means:
$$ \begin{align} \operatorname{lcm} (\operatorname{lcm} (a, b), c) &= {p_1}^{\max \{x_1, y_1, z_1\} } {p_2}^{\max \{x_2,y_2, z_2\} } {p_3}^{\max \{x_3,y_3, z_3\} } \ldots {p_n}^{\max \{x_n, y_n, z_n\} } \\ &= \operatorname{lcm} (a, b, c) \end{align}$$