reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem Th12:
  a>0 & n>=1 or a<>0 & n is odd implies n-root (1/a) = 1/(n-root a)
proof
  assume
A1: a>0 & n>=1 or a<>0 & n is odd;
A2: now
    let a,n;
    assume
A3: a>0 & n>=1;
    hence n-root (1/a) = n -Root (1/a) by Def1
      .= 1/(n -Root a) by A3,PREPOWER:23
      .= 1/(n-root a) by A3,Def1;
  end;
 now
    assume that
A4: a<>0 and
A5: n is odd;
A6: n>=1 by A5,ABIAN:12;
 now per cases by A4;
      suppose
     a>0;
        hence thesis by A2,A6;
      end;
      suppose
    a<0;
then A7:    -a>0 by XREAL_1:58;
        thus
        1/(n-root a) = 1/(-(n-root (-a))) by A5,Th10
          .= - 1/(n-root (-a)) by XCMPLX_1:188
          .= - n-root (1/(-a)) by A2,A6,A7
          .= - n-root (-1/a) by XCMPLX_1:188
          .= n-root (1/a) by A5,Th10;
      end;
    end;
    hence thesis;
  end;
  hence thesis by A1,A2;
end;
