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Proposition: Let a and a' be in the same residue class and also b and b' be in the same residue class. The result of any integer operation involving a and b is in the same residue class as if the operation were performed with a' and b' instead:
forall m in Z>0, a in Z, a' in Z, b in Z, b' in Z: [a]m = [a']m /\ [b]m = [b']m => [a+b]m = [a'+b']m /\ [-a]m = [-a']m /\ [a*b]m = [a'*b']m.