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 Th10:
  n is odd implies n-root a = - n-root (-a)
proof
  assume
A1: n is odd;
    then
A2: ex m st n = 2*m + 1 by ABIAN:9;
A3: n>=1 by A1,ABIAN:12;
 now per cases;
    suppose
A4:   a<0;
      thus
      then n-root a = - n -Root (-a) by A2,Def1
        .= - n-root (-a) by A3,A4,Def1;
    end;
    suppose
A5:   a=0;
      hence n-root a = 0 by A3,Th5
        .= - n-root (-a) by A3,A5,Th5;
    end;
    suppose
A6:   a>0;
then    -a<-0 by XREAL_1:24;
      hence - n-root (-a) = -(- n -Root (-(-a))) by A2,Def1
        .= n-root a by A3,A6,Def1;
    end;
  end;
  hence thesis;
end;
