reserve a, b, p, q for Real;

theorem
  for p, q be Real st 0 < p & 0 < q
  for a be Real st 0 <= a holds
  (a to_power p) * (a to_power q) = a to_power (p+q)
proof
  let p, q be Real such that
A1: 0 < p and
A2: 0 < q;
  let a be Real such that
A3: 0 <=a;
  now
    per cases;
    case
A4:   a=0;
      then (a to_power p) * (a to_power q) = 0 * (0 to_power q) by A1,
POWER:def 2
        .= 0;
      hence thesis by A1,A2,A4,POWER:def 2;
    end;
    case
      a <> 0;
      hence thesis by A3,POWER:27;
    end;
  end;
  hence thesis;
end;
