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Theorem. The product of the additive inverse of the multiplicative identity with itself is equal to the multiplicative identity.

Proof. The sum of the multiplicative identity and its additive inverse is the additive identity: that is, the expression "1 + (–1)" is equal to the expression "0". Multiplying both of these expressions by the additive inverse of the multiplicative identity, then applying the distributivity axiom, the theorem of multiplication by the additive identity, and the law of multiplicative identity, we get:

–1(–1 + 1) = –1(0)

(–1)(–1) + (–1)1 = 0

(–1)(–1) + (–1) = 0

But then adding the multiplicative identity to both of these expressions and applying the law of additive inverses and the law of additive identity, we get:

(–1)(–1) + (–1) + 1 = 0 + 1

(–1)(–1) = 1

But that's what I've been trying to tell you this whole time.

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