Proof: Additive Inverse 2

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	(a ⋅ 2) + ((a ⋅ 2) ⋅ (-1)) = 0	(a ⋅ 2) + ((a ⋅ 2) ⋅ (-1)) = 0 (Additive Inverse Property)
2	(a ⋅ 2) ⋅ (-1) = a ⋅ (2 ⋅ (-1))	(a ⋅ 2) ⋅ (-1) = a ⋅ (2 ⋅ (-1)) (Associative Property of Multiplication)
3	2 ⋅ (-1) = -2	2 ⋅ (-1) = -2 (Fixed Value Statement)
4	a ⋅ (2 ⋅ (-1)) = a ⋅ (-2)	if 2 ⋅ (-1) = -2, then a ⋅ (2 ⋅ (-1)) = a ⋅ (-2) (Multiplicative Property of Equality Variation 1)
5	(a ⋅ 2) ⋅ (-1) = a ⋅ (-2)	if a ⋅ (2 ⋅ (-1)) = a ⋅ (-2) and (a ⋅ 2) ⋅ (-1) = a ⋅ (2 ⋅ (-1)), then (a ⋅ 2) ⋅ (-1) = a ⋅ (-2) (Transitive Property of Equality)
6	(a ⋅ 2) + (a ⋅ (-2)) = 0	if (a ⋅ 2) ⋅ (-1) = a ⋅ (-2) and (a ⋅ 2) + ((a ⋅ 2) ⋅ (-1)) = 0, then (a ⋅ 2) + (a ⋅ (-2)) = 0 (Substitution Example 10)

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