Establishing equality of the various decompositions of mulpos₇ (p, q, r, s, t, u, v) is simple in principle, but the notation is rather unwieldy.

• All expressions in the "usual notation" column below, which are steps in the derivation, are equal.
• The "concise notation" column leaves out the mulpos_ns and commas for what may be a clearer representation.
• The "justification" column explains how each step was derived from the previous.

Each step of the derivation, which involves no more than adding or removing parentheses, uses one of these identities:

mulpos5 (p, q, r, s, t) = mulpos3 (mulpos3 (p, q, r), s, t)		#I
mulpos5 (p, q, r, s, t) = mulpos3 (p, mulpos3 (q, r, s), t)		#II
mulpos5 (p, q, r, s, t) = mulpos3 (p, q, mulpos3 (r, s, t))		#III

Usual notation		Concise notation	Justification
#IV	mulpos5 (mulpos3 (p, q, r), s, t, u, v)	( ( p q r ) s t u v )
	mulpos3 (mulpos3 (mulpos3 (p, q, r), s, t), u, v)	( ( ( p q r ) s t ) u v )	add () to ( p q r ) s t	#I
#IX	mulpos3 (mulpos5 (p, q, r, s, t), u, v)	( ( p q r s t ) u v )	remove () from ( p q r )	#I
	mulpos3 (mulpos3 (p, mulpos3 (q, r, s), t), u, v)	( ( p ( q r s ) t ) u v )	add () to q r s	#II
#V	mulpos5 (p, mulpos3 (q, r, s), t, u, v)	( p ( q r s ) t u v )	remove () from ( p ( q r s ) t )	#I
	mulpos3 (p, mulpos3 (mulpos3 (q, r, s), t, u), v)	( p ( ( q r s ) t u ) v )	add () to ( q r s ) t u	#II
#X	mulpos3 (p, mulpos5 (q, r, s, t, u), v)	( p ( q r s t u ) v )	remove () from ( q r s )	#I
	mulpos3 (p, mulpos3 (q, mulpos3 (r, s, t), u), v)	( p ( q ( r s t ) u ) v )	add () to r s t	#II
#VI	mulpos5 (p, q, mulpos3 (r, s, t), u, v)	( p q ( r s t ) u v )	remove () from ( q ( r s t ) u )	#II
	mulpos3 (p, q, mulpos3 (mulpos3 (r, s, t), u, v))	( p q ( ( r s t ) u v ) )	add () to ( r s t ) u v	#III
#XI	mulpos3 (p, q, mulpos5 (r, s, t, u, v))	( p q ( r s t u v ) )	remove () from ( r s t )	#I
	mulpos3 (p, q, mulpos3 (r, mulpos3 (s, t, u), v))	( p q ( r ( s t u ) v ) )	add () to s t u	#II
#VII	mulpos5 (p, q, r, mulpos3 (s, t, u), v)	( p q r ( s t u ) v )	remove () from ( r ( s t u ) v )	#III
	mulpos3 (p, q, mulpos3 (r, mulpos3 (s, t, u), v))	( p q ( r ( s t u ) v ) )	add () to r ( s t u ) v	#III
#XI	mulpos3 (p, q, mulpos5 (r, s, t, u, v))	( p q ( r s t u v ) )	remove () from ( s t u )	#II
	mulpos3 (p, q, mulpos3 (r, s, mulpos3 (t, u, v)))	( p q ( r s ( t u v ) ) )	add () to t u v	#III
#VIII	mulpos5 (p, q, r, s, mulpos3 (t, u, v))	( p q r s ( t u v ) )	remove () from ( r s ( t u v ) )	#III