Let's prove the following theorem:

if not (x^m = 0), then x^{((-1) ⋅ m)} = 1 / (x^m)

Power to a negative number is 1 divided by the power to the inverse of the number. For example, 2^-3 = 1 / 2³

Given

1	not (x^m = 0)

Proof Table

#	Claim	Reason
1	x^{(m + ((-1) ⋅ m))} = (x^m) ⋅ (x^{((-1) ⋅ m)})	x^{(m + ((-1) ⋅ m))} = (x^m) ⋅ (x^{((-1) ⋅ m)}) (Power By M and N)
2	m + ((-1) ⋅ m) = 0	m + ((-1) ⋅ m) = 0 (additive_inverse_2)
3	x^{(m + ((-1) ⋅ m))} = x⁰	if m + ((-1) ⋅ m) = 0, then x^{(m + ((-1) ⋅ m))} = x⁰ (Substitution)
4	x⁰ = 1	x⁰ = 1 (Power By 0)
5	x^{(m + ((-1) ⋅ m))} = 1	if x^{(m + ((-1) ⋅ m))} = x⁰ and x⁰ = 1, then x^{(m + ((-1) ⋅ m))} = 1 (Transitive Property of Equality)
6	1 = (x^m) ⋅ (x^{((-1) ⋅ m)})	if x^{(m + ((-1) ⋅ m))} = 1 and x^{(m + ((-1) ⋅ m))} = (x^m) ⋅ (x^{((-1) ⋅ m)}), then 1 = (x^m) ⋅ (x^{((-1) ⋅ m)}) (Transitive Property of Equality Variation 2)
7	x^{((-1) ⋅ m)} = 1 / (x^m)	if 1 = (x^m) ⋅ (x^{((-1) ⋅ m)}) and not (x^m = 0), then x^{((-1) ⋅ m)} = 1 / (x^m) (Divide by Term)

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