Multiplicative Identity 2
Commutative Property Variation 1
Substitution 2
Distributive Property 4
Multiplicative Property of Equality Variation 1
Addition Theorem
Add Three
Add Four
Transitive Property of Equality Variation 2
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
Divide Each Side
Reduce Addition

Proof: Add Four

Let's prove the following theorem:

((a + a) + a) + a = a ⋅ 4

Proof:

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Proof Table
# Claim Reason
1 (a + a) + a = a ⋅ 3 (a + a) + a = a ⋅ 3
2 ((a + a) + a) + a = (a ⋅ 3) + a if (a + a) + a = a ⋅ 3, then ((a + a) + a) + a = (a ⋅ 3) + a
3 a = a ⋅ 1 a = a ⋅ 1
4 (a ⋅ 3) + a = (a ⋅ 3) + (a ⋅ 1) if a = a ⋅ 1, then (a ⋅ 3) + a = (a ⋅ 3) + (a ⋅ 1)
5 (a ⋅ 3) + (a ⋅ 1) = a ⋅ (3 + 1) (a ⋅ 3) + (a ⋅ 1) = a ⋅ (3 + 1)
6 3 + 1 = 4 3 + 1 = 4
7 a ⋅ (3 + 1) = a ⋅ 4 if 3 + 1 = 4, then a ⋅ (3 + 1) = a ⋅ 4
8 (a ⋅ 3) + (a ⋅ 1) = a ⋅ 4 if (a ⋅ 3) + (a ⋅ 1) = a ⋅ (3 + 1) and a ⋅ (3 + 1) = a ⋅ 4, then (a ⋅ 3) + (a ⋅ 1) = a ⋅ 4
9 (a ⋅ 3) + a = a ⋅ 4 if (a ⋅ 3) + a = (a ⋅ 3) + (a ⋅ 1) and (a ⋅ 3) + (a ⋅ 1) = a ⋅ 4, then (a ⋅ 3) + a = a ⋅ 4
10 ((a + a) + a) + a = a ⋅ 4 if ((a + a) + a) + a = (a ⋅ 3) + a and (a ⋅ 3) + a = a ⋅ 4, then ((a + a) + a) + a = a ⋅ 4
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