reserve k for Element of NAT;
reserve r,r1 for Real;
reserve i for Integer;
reserve q for Rational;

theorem Th9:
  ex n being Nat st -n <= r
proof
 [\ r /] is Element of INT by INT_1:def 2;
  then A1: [\
 r /] <= r & ex k being Nat st ( [\ r /] = k or [\ r /] = -k)
  by INT_1:def 1,def 6;
  per cases;
  suppose
 [\ r /] < 0;
    hence thesis by A1;
  end;
  suppose
A2: [\ r /] >= 0;
    take 0;
    thus thesis by A2,INT_1:def 6;
  end;
end;
