Distributive Property 2

Equality Example

Distributive Property 3

Associative Property of Multiplication 2

Transitive Property Application 1

Substitution 10

Transitive Property of Equality Variation 1

Divide Both Sides

Multiplicative Property of Equality Variation 2

Transitive Property of Equality Variation 3

Division is Commutative

Associative Property

Multiplicative Property of Equality Variation 1

Substitution 12

Transitive Property of Equality Variation 2

Divide Simplify 2

Proof: Simplify 3

Let's prove the following theorem:

if not (2 = 0), then ((b ⋅ 2) + (a ⋅ 2)) / 2 = b + a

Proof:

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Given

1	not (2 = 0)

Proof Table

#	Claim	Reason
1	((b ⋅ 2) + (a ⋅ 2)) / 2 = ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2)	((b ⋅ 2) + (a ⋅ 2)) / 2 = ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) (Division is Multiplication by Inverse)
2	((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = ((b ⋅ 2) ⋅ (1 / 2)) + ((a ⋅ 2) ⋅ (1 / 2))	((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = ((b ⋅ 2) ⋅ (1 / 2)) + ((a ⋅ 2) ⋅ (1 / 2)) (Distributive Property Variation 3)
3	(b ⋅ 2) ⋅ (1 / 2) = b	if not (2 = 0), then (b ⋅ 2) ⋅ (1 / 2) = b (Simplify 2)
4	(a ⋅ 2) ⋅ (1 / 2) = a	if not (2 = 0), then (a ⋅ 2) ⋅ (1 / 2) = a (Simplify 2)
5	((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = b + ((a ⋅ 2) ⋅ (1 / 2))	if ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = ((b ⋅ 2) ⋅ (1 / 2)) + ((a ⋅ 2) ⋅ (1 / 2)) and (b ⋅ 2) ⋅ (1 / 2) = b, then ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = b + ((a ⋅ 2) ⋅ (1 / 2)) (Substitution 6)
6	((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = b + a	if ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = b + ((a ⋅ 2) ⋅ (1 / 2)) and (a ⋅ 2) ⋅ (1 / 2) = a, then ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = b + a (Substitution 2)
7	((b ⋅ 2) + (a ⋅ 2)) / 2 = b + a	if ((b ⋅ 2) + (a ⋅ 2)) / 2 = ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) and ((b ⋅ 2) + (a ⋅ 2)) ⋅ (1 / 2) = b + a, then ((b ⋅ 2) + (a ⋅ 2)) / 2 = b + a (Transitive Property of Equality)

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