Cancellation Law for Addition PROOF (real not fake) Then, a+b=a+c. a+b+(−a)=a+c+(−a) ; Axioms: APE a+(b+(−a))=a+(c+(−a)) ; Axioms: Associativity for Addition a+((−a)+b)=(a+(−a))+c ; Axioms: Commutativity (a+(−a))+b=(a+(−a))+c ; Axioms: Associativity 0+b=0+c ; Axioms: Ex. of Additive Inverse b=c ; Axioms: Ex. of Additive Identity