For our proof, we will use this property:
$$ \max(a,b,c)=a+b+c-\min(a,b)-\min(a,c)-\min(b,c) + \min(a,b,c) $$
According to the fundamental theorem of arithmetic:
$$ \begin{align} a &= {p_1}^{x_1} {p_2}^{x_2} {p_3}^{x_3} \cdots {p_n}^{x_n} \\ b &= {p_1}^{y_1} {p_2}^{y_2} {p_3}^{y_3} \cdots {p_n}^{y_n} \\ c &= {p_1}^{z_1} {p_2}^{z_2} {p_3}^{z_3} \cdots {p_n}^{z_n} \\ (a,b) &= {p_1}^{\min(x_1,y_1)} {p_2}^{\min(x_2,y_2)} {p_3}^{\min(x_3,y_3)} \cdots{p_n}^{\min(x_n,y_n)} \\ (a,c) &= {p_1}^{\min(x_1,z_1)} {p_2}^{\min(x_2,z_2)} {p_3}^{\min(x_3,z_3)} \cdots{p_n}^{\min(x_n,z_n)} \\ (b,c) &= {p_1}^{\min(y_1,z_1)} {p_2}^{\min(y_2,z_2)} {p_3}^{\min(y_3,z_3)} \cdots {p_n}^{\min(y_n,z_n)} \end{align} $$
Also:
$$ \begin{align} abc &= {p_1}^{x_1+y_1+z_1} {p_2}^{x_2+y_2+z_2} {p_3}^{x_3+y_3+z_3} \cdots {p_n}^{x_n+y_n+z_n} \\ (a,b,c) &= {p_1}^{\min(x_1,y_1,z_1)} {p_2}^{\min(x_2,y_2,z_2)} {p_3}^{\min(x_3,y_3,z_3)} \cdots{p_n}^{\min(x_n,y_n,z_n)} \\ [a,b,c] &= {p_1}^{\max(x_1,y_1,z_1)} {p_2}^{\max(x_2,y_2,z_2)} {p_3}^{\max(x_3,y_3,z_3)} \cdots{p_n}^{\max(x_n,y_n,z_n)} \end{align} $$
If we multiply \(abc\) and \((a,b,c)\):
$$\begin{align} abc(a,b,c) = & \ ({p_1}^{x_1+y_1+z_1 } {p_2}^{x_2+y_2+z_2} {p_3}^{x_3+y_3+z_3} \cdots {p_n}^{x_n+y_n+z_n}) *\\ & ({p_1}^{\min(x_1,y_1,z_1)} {p_2}^{\min(x_2,y_2,z_2)} {p_3}^{\min(x_3,y_3,z_3)} \cdots{p_n}^{\min(x_n,y_n,z_n)}) \\ = & \ {p_1}^{x_1+y_1+z_1 +\min(x_1+y_1+z_1)} {p_2}^{x_2+y_2+z_2+\min(x_2+y_2+z_2)} \cdots {p_n}^{x_n+y_n+z_n+\min(x_n+y_n+z_n)} \end{align}$$
If we multiply \((a,b)\), \((a,c)\) and \((b,c)\):
$$(a,b)(a,c)(b,c) = {p_1}^{\min(x_1,y_1)+\min(x_1,z_1)+\min(y_1,z_1)} {p_2}^{\min(x_2,y_2)+\min(x_2,z_2)+\min(y_2,z_2)} \cdots {p_n}^{\min(x_n,y_n)+\min(x_n,z_n)+\min(y_n,z_n)} $$
Now we do \(abc(a,b,c)/(a,b)(a,c)(b,c)\):
$$\frac{abc(a,b,c)}{(a,b)(a,c)(b,c)} = \frac{{p_1}^{x_1+y_1+z_1 +\min(x_1+y_1+z_1)} {p_2}^{x_2+y_2+z_2+\min(x_2+y_2+z_2)} \cdots {p_n}^{x_n+y_n+z_n+\min(x_n+y_n+z_n)}}{{p_1}^{\min(x_1,y_1)+\min(x_1,z_1)+\min(y_1,z_1)} {p_2}^{\min(x_2,y_2)+\min(x_2,z_2)+\min(y_2,z_2)} \cdots {p_n}^{\min(x_n,y_n)+\min(x_n,z_n)+\min(y_n,z_n)}} $$
Simplifying:
$$\begin{align} \frac{abc(a,b,c)}{(a,b)(a,c)(b,c)} = & \ {p_1}^{x_1+y_1+z_1 +\min(x_1+y_1+z_1)-\min(x_1,y_1)-\min(x_1,z_1)-\min(y_1,z_1)} * \\ & \ {p_2}^{x_2+y_2+z_2 +\min(x_2+y_2+z_2)-\min(x_2,y_2)-\min(x_2,z_2)-\min(y_2,z_2)} * \cdots * \\ & \ {p_n}^{x_n+y_n+z_n +\min(x_n+y_n+z_n)-\min(x_n,y_n)-\min(x_n,z_n)-\min(y_n,z_n)} \end{align}$$
Using the property mentioned above:
$$ \frac{abc(a,b,c)}{(a,b)(a,c)(b,c)} = {p_1}^{\max(x_1,y_1,z_1)} * {p_2}^{\max(x_2,y_2,z_2)} * \cdots * {p_n}^{\max(x_n,y_n,z_n)} $$
This is the same as \([a,b,c]\).