Multiplicative Property of Equality Variation 1

Substitution 12

Substitution 15

Associative Property of Multiplication 2

Transitive Property of Equality Variation 2

Transitive Property Application 2

Substitution 10

Reorder Terms 7

Transitive Property of Equality Variation 1

Proof: Algebra 8

Let's prove the following theorem:

(s / 2) ⋅ (s / 2) = (s ⋅ s) / 4

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	(s / 2) ⋅ (s / 2) = (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2))	if s / 2 = s ⋅ (1 / 2), then (s / 2) ⋅ (s / 2) = (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) (Substitution 15)
3	(s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2)	(s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) (associative_property_of_multiplication_2)
4	((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2)	((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) (Swap Terms 2 and 3)
5	(s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2)	if (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) and ((s ⋅ (1 / 2)) ⋅ s) ⋅ (1 / 2) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2), then (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) (Transitive Property of Equality)
6	(1 / 2) ⋅ (1 / 2) = 1 / 4	(1 / 2) ⋅ (1 / 2) = 1 / 4 (Fixed Value Statement)
7	((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ ((1 / 2) ⋅ (1 / 2))	((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ ((1 / 2) ⋅ (1 / 2)) (Associative Property of Multiplication)
8	((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ (1 / 4)	if ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ ((1 / 2) ⋅ (1 / 2)) and (1 / 2) ⋅ (1 / 2) = 1 / 4, then ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ (1 / 4) (Substitution 12)
9	(s ⋅ s) / 4 = (s ⋅ s) ⋅ (1 / 4)	(s ⋅ s) / 4 = (s ⋅ s) ⋅ (1 / 4) (Division is Multiplication by Inverse)
10	((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) / 4	if ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) ⋅ (1 / 4) and (s ⋅ s) / 4 = (s ⋅ s) ⋅ (1 / 4), then ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) / 4 (Transitive Property of Equality Variation 1)
11	(s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = (s ⋅ s) / 4	if (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) and ((s ⋅ s) ⋅ (1 / 2)) ⋅ (1 / 2) = (s ⋅ s) / 4, then (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = (s ⋅ s) / 4 (Transitive Property of Equality)
12	(s / 2) ⋅ (s / 2) = (s ⋅ s) / 4	if (s / 2) ⋅ (s / 2) = (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) and (s ⋅ (1 / 2)) ⋅ (s ⋅ (1 / 2)) = (s ⋅ s) / 4, then (s / 2) ⋅ (s / 2) = (s ⋅ s) / 4 (Transitive Property of Equality)

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