Proof: Algebra 16

Let's prove the following theorem:

(a ⋅ (s / 2)) / s = a / 2

Proof:

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Proof Table

#	Claim	Reason
1	s / 2 = s ⋅ (1 / 2)	s / 2 = s ⋅ (1 / 2) (Division is Multiplication by Inverse)
2	a ⋅ (s / 2) = a ⋅ (s ⋅ (1 / 2))	if s / 2 = s ⋅ (1 / 2), then a ⋅ (s / 2) = a ⋅ (s ⋅ (1 / 2)) (Multiplicative Property of Equality Variation 1)
3	a ⋅ (s ⋅ (1 / 2)) = (a ⋅ s) ⋅ (1 / 2)	a ⋅ (s ⋅ (1 / 2)) = (a ⋅ s) ⋅ (1 / 2) (associative_property_of_multiplication_2)
4	a ⋅ (s / 2) = (a ⋅ s) ⋅ (1 / 2)	if a ⋅ (s / 2) = a ⋅ (s ⋅ (1 / 2)) and a ⋅ (s ⋅ (1 / 2)) = (a ⋅ s) ⋅ (1 / 2), then a ⋅ (s / 2) = (a ⋅ s) ⋅ (1 / 2) (Transitive Property of Equality)
5	(a ⋅ (s / 2)) / s = ((a ⋅ s) ⋅ (1 / 2)) / s	if a ⋅ (s / 2) = (a ⋅ s) ⋅ (1 / 2), then (a ⋅ (s / 2)) / s = ((a ⋅ s) ⋅ (1 / 2)) / s (Divide Both Sides)
6	((a ⋅ s) ⋅ (1 / 2)) / s = ((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s)	((a ⋅ s) ⋅ (1 / 2)) / s = ((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) (Division is Multiplication by Inverse)
7	((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) = ((a ⋅ s) ⋅ (1 / s)) ⋅ (1 / 2)	((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) = ((a ⋅ s) ⋅ (1 / s)) ⋅ (1 / 2) (Swap 2 and 3 Theorem)
8	(a ⋅ s) ⋅ (1 / s) = a ⋅ 1	(a ⋅ s) ⋅ (1 / s) = a ⋅ 1 (divide_simplify_2)
9	a ⋅ 1 = a	a ⋅ 1 = a (Multiplicative Identity)
10	(a ⋅ s) ⋅ (1 / s) = a	if (a ⋅ s) ⋅ (1 / s) = a ⋅ 1 and a ⋅ 1 = a, then (a ⋅ s) ⋅ (1 / s) = a (Transitive Property of Equality)
11	((a ⋅ s) ⋅ (1 / s)) ⋅ (1 / 2) = a ⋅ (1 / 2)	if (a ⋅ s) ⋅ (1 / s) = a, then ((a ⋅ s) ⋅ (1 / s)) ⋅ (1 / 2) = a ⋅ (1 / 2) (Multiplicative Property of Equality)
12	((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) = a ⋅ (1 / 2)	if ((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) = ((a ⋅ s) ⋅ (1 / s)) ⋅ (1 / 2) and ((a ⋅ s) ⋅ (1 / s)) ⋅ (1 / 2) = a ⋅ (1 / 2), then ((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) = a ⋅ (1 / 2) (Transitive Property of Equality)
13	((a ⋅ s) ⋅ (1 / 2)) / s = a ⋅ (1 / 2)	if ((a ⋅ s) ⋅ (1 / 2)) / s = ((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) and ((a ⋅ s) ⋅ (1 / 2)) ⋅ (1 / s) = a ⋅ (1 / 2), then ((a ⋅ s) ⋅ (1 / 2)) / s = a ⋅ (1 / 2) (Transitive Property of Equality)
14	(a ⋅ (s / 2)) / s = a ⋅ (1 / 2)	if (a ⋅ (s / 2)) / s = ((a ⋅ s) ⋅ (1 / 2)) / s and ((a ⋅ s) ⋅ (1 / 2)) / s = a ⋅ (1 / 2), then (a ⋅ (s / 2)) / s = a ⋅ (1 / 2) (Transitive Property of Equality)
15	a ⋅ (1 / 2) = a / 2	a ⋅ (1 / 2) = a / 2 (division_theorem)
16	(a ⋅ (s / 2)) / s = a / 2	if (a ⋅ (s / 2)) / s = a ⋅ (1 / 2) and a ⋅ (1 / 2) = a / 2, then (a ⋅ (s / 2)) / s = a / 2 (Transitive Property of Equality)

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