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