Definition powerRZ (x:R) (n:Z) :=
match n with
| Z0 => 1
| Zpos p => x ^ Pos.to_nat p
| Zneg p => / x ^ Pos.to_nat p
end.
Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope.

Lemma powerRZ_pos_sub (x:R) (n m:positive) : x <> 0 ->
x ^Z (Z.pos_sub n m) = x ^ Pos.to_nat n * / x ^ Pos.to_nat m.