Let's prove the following theorem:

if not (a = 0), then ((1 / a) ⋅ a) ⋅ x = x

Given

1	not (a = 0)

Proof Table

#	Claim	Reason
1	(1 / a) ⋅ a = 1	if not (a = 0), then (1 / a) ⋅ a = 1 (Inverse Product Theorem)
2	((1 / a) ⋅ a) ⋅ x = 1 ⋅ x	if (1 / a) ⋅ a = 1, then ((1 / a) ⋅ a) ⋅ x = 1 ⋅ x (Multiplicative Property of Equality)
3	1 ⋅ x = x	1 ⋅ x = x (multiplicative_identity_3)
4	((1 / a) ⋅ a) ⋅ x = x	if 1 ⋅ x = x and ((1 / a) ⋅ a) ⋅ x = 1 ⋅ x, then ((1 / a) ⋅ a) ⋅ x = x (Transitive Property of Equality)

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