[Lemma] ∀p,q,r∈ℕ×ℕ ( p ~ q ⇒ p·r ~ q·r )

Given p,q,r ∈ ℕ×ℕ
	If	p ~ q
		Let p = (a,b)
		Let q = (c,d)
		Let r = (x,y)

		a+d = b+c
		p.r = (a.x + b.y , a.y + b.x)
		q.r = (c.x + d.y , c.y + d.x)

		a.x + d.x = b.x + c.x
		a.y + d.y = b.y + c.y

		a.x + b.y + c.y + d.x
		= a.x + d.x + b.y + c.y
		= b.x + c.x + a.y + d.y
		= a.y + b.x + c.x + d.y

		p.r ~ q.r

[Lemma] ∀p,q ∈ ℕ×ℕ (p.q ~ q.p)

Given p,q ∈ ℕ×ℕ
	Let p = (a,b)
	Let q = (c,d)

	p.q = (ac + bd , ad + bc)
	q.p = (ca + db , cb + da) = (ac + bd , ad + bc) = p.q