Proof: Add Four

Let's prove the following theorem:

Proof:

Proof Table

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

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