Proof: Add Three

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	a + a = a ⋅ 2	a + a = a ⋅ 2 (addition_theorem)
2	a = a ⋅ 1	a = a ⋅ 1 (multiplicative_identity_2)
3	(a + a) + a = (a ⋅ 2) + a	if a + a = a ⋅ 2, then (a + a) + a = (a ⋅ 2) + a (Additive Property of Equality)
4	(a ⋅ 2) + a = (a ⋅ 2) + (a ⋅ 1)	if a = a ⋅ 1, then (a ⋅ 2) + a = (a ⋅ 2) + (a ⋅ 1) (Substitution)
5	(a ⋅ 2) + (a ⋅ 1) = a ⋅ (2 + 1)	(a ⋅ 2) + (a ⋅ 1) = a ⋅ (2 + 1) (Distributive Property Variation 4)
6	(a ⋅ 2) + a = a ⋅ (2 + 1)	if (a ⋅ 2) + a = (a ⋅ 2) + (a ⋅ 1) and (a ⋅ 2) + (a ⋅ 1) = a ⋅ (2 + 1), then (a ⋅ 2) + a = a ⋅ (2 + 1) (Transitive Property of Equality)
7	2 + 1 = 3	2 + 1 = 3 (Fixed Value Statement)
8	a ⋅ (2 + 1) = a ⋅ 3	if 2 + 1 = 3, then a ⋅ (2 + 1) = a ⋅ 3 (Multiplicative Property of Equality Variation 1)
9	(a ⋅ 2) + a = a ⋅ 3	if (a ⋅ 2) + a = a ⋅ (2 + 1) and a ⋅ (2 + 1) = a ⋅ 3, then (a ⋅ 2) + a = a ⋅ 3 (Transitive Property of Equality)
10	(a + a) + a = a ⋅ 3	if (a + a) + a = (a ⋅ 2) + a and (a ⋅ 2) + a = a ⋅ 3, then (a + a) + a = a ⋅ 3 (Transitive Property of Equality)

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