Question

fill up a^m ÷a^n+a ---- where m and n are natural numbers please give the answer soon

Answer 1

If you are using Peano axioms, induct on n with m fixed.

Note: n+ is the sucessor of n.

Axiom 1. 0∈N.

Axiom 2. if n∈N then n+∈N.

Definition 3 (Adition). m+0:=m and m+n+:=(m+n)+.

Proof. Induct on n (keeping m fixed). Consider the case base n=0. Since by hypothesis we have m∈N, then m+0=m∈N by Definition 3. Now suppose inductively that m+n∈N; we now have to show that m+n+∈N. By Definition 3, we have m+n+=(m+n)+. By Axiom 2, since m+n∈N, we conclude (m+n)+=m+n+∈N. Thus m+n∈N for every m,n∈N. This close the induction.