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Proposition: Let Z' denote the old construction of the integers and Z denote the new one. The function i: Z' -> Z
is an isomorphism with respect to 0, +, -, *, <, i.e., i is bijective and for all x in Z' and y in Z', we have:
i(x) := [x]
i(0Z') = 0Z, i(x+Z'y) = i(x)+Zi(y), i(-Z' x) = -Z i(x), i(x-Z'y) = i(x)-Zi(y), ...
Inverse Isomorphism: j: Z -> Z', j(x) := I(x)