We define the power of a relation in the following way:
$R^{1}=R$
$R^{2}=R \circ R$
$R^{3}=R \circ R^{2}$
and we want that for every $n,m: R^{n} \circ R^{m}=R^{n+m} $.
More formally we can also define it by induction in the following way:
($R^{0}=Id_A$)
$R^{1}=R$
$R^{n+1}=R^{n}\circ R$
Given that definition I would like to prove that for every set $A$, a relation $R$ and a natural number $n$: $R^{0}\circ R^{n}=R^n$ implies $R^{0}=Id_A$.