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
  1-root a = a
proof
 now per cases;
    suppose
A1:   a>=0;
      hence 1-root a = 1 -Root a by Def1
        .= a by A1,PREPOWER:21;
    end;
    suppose
A2:   a<0;
   1 = 2*0 + 1;
      hence 1-root a = - 1 -Root (-a) by A2,Def1
        .= -(-a) by A2,PREPOWER:21
        .= a;
    end;
  end;
  hence thesis;
end;
