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) to_power q = a to_power (p*q)
proof
  let p, q be Real such that
A1: 0 < p and
A2: 0 < q;
A3: 0 < p*q by A1,A2,XREAL_1:129;
  let a be Real such that
A4: 0 <=a;
  now
    per cases;
    case
A5:   a=0;
      then (a to_power p) to_power q = 0 to_power q by A1,POWER:def 2
        .= 0 by A2,POWER:def 2;
      hence thesis by A3,A5,POWER:def 2;
    end;
    case
      a <> 0;
      hence thesis by A4,POWER:33;
    end;
  end;
  hence thesis;
end;
