Factorial as Product of Three Factorials

Theorem
From Factorial as Product of Two Factorials:
 * $\left({n!}\right)! = n! \left({n! - 1}\right)!$

Apart from the general pattern, following directly from the definition of the factorial and the above quoted result:
 * $\left({\left({n!}\right)!}\right)! = n! \left({n! - 1}\right)! \left({\left({n!}\right)! - 1}\right)!$

The only known factorial which is the product of three factorials is:
 * $10! = 7! \, 5! \, 3!$