theorem Th45:
  a>=0 & n>=1 implies a to_power (1/n) = n-root a
proof
  assume that
A1: a>=0 and
A2: n>=1;
  reconsider p=n" as Rational;
 now per cases by A1;
    suppose
   a>0;
      hence a to_power (1/n) = a #Q p by Th44
        .= n -Root a by A2,PREPOWER:50;
    end;
    suppose
A3:   a=0;
      hence a to_power (1/n) = 0 by A2,Def2
        .= n -Root a
      by A2,A3,PREPOWER:def 2;
    end;
  end;
  hence thesis by A1,A2,Def1;
end;
