reserve n, k, r, m, i, j for Nat;

theorem Th10:
  for a being non zero Real holds (a to_power k) * (a
  to_power (-k)) = 1
proof
  let a be non zero Real;
  (a to_power k) * (a to_power (-k)) = (a #Z k) * (a to_power (-k)) by
POWER:def 2
    .= (a #Z k) * (a #Z (-k)) by POWER:def 2
    .= a #Z (k + (-k)) by PREPOWER:44
    .= 1 by PREPOWER:34;
  hence thesis;
end;
