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

theorem Th5:
  for a being Real holds a to_power (k+m) = (a to_power k) *
  (a to_power m)
proof
  let a be Real;
  thus a to_power (k+m) = a |^ (k+m) by POWER:41
    .= a |^ k * a |^ m by NEWTON:8
    .= (a to_power k) * a |^ m by POWER:41
    .= (a to_power k) * (a to_power m) by POWER:41;
end;
