
theorem
  for n,m being Nat holds 0 to_power (n+m) = (0 to_power n)*(0 to_power m)
proof
  let n,m be Nat;
  per cases;
  suppose
A1: n > 0 or m > 0;
    then 0 to_power n = 0 or 0 to_power m = 0 by POWER:def 2;
    hence thesis by A1,POWER:def 2;
  end;
  suppose
A2: n = 0 & m = 0;
    then 0 to_power (n+m) = 1 by POWER:24;
    hence thesis by A2;
  end;
end;
