User blog comment:LittlePeng9/Addition is commutative/@comment-2033667-20141002052007

okay so \(\forall x \forall y : x = y \Rightarrow S(x) = S(y)\) is given both times with simply the inference rule "S." i think you mean \(A_4\), since all other Peano axioms are given by their numbers

no deduction is given for \(a = 0 + a\) in its leftmost appearance (the halfway point correctly derives it from A3 and universal elimination)