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Proposition: For every integer x and y the difference is defined:
forall x in Z, y in Z: x = (x-y)+y.
Proof: Take arbitrary x in Z and y in Z. We have
(x-y)+y = (definition of -) (x+(-y))+y = (associativity of +) x+((-y+y)) = (*) x+0 = (definition of + and 0) x.