Negative operation Neg(x)
Let’s define unary operation Neg : Z → Z as follows:
∀x ∈ Z Neg(x) := 0 – x
Thus
Neg(0) = 0
THEOREM 2 ∀x y ∈ Z Neg(x-y) = y-x
PROOF
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END of PROOF
As a corollary we get:
THEOREM 3 ∀x ∈ Z Neg(Neg(x)) = x
PROOF
Neg(Neg(x)) = Neg(0-x) = x-0 = x
END of PROOF