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

theorem Th2:
  INT=INT- \/ NAT
proof
 for x being object st x in INT holds x in INT- \/ NAT
  proof
    let x be object;
    assume x in INT;
    then consider k being Nat such that
A1: x = k or x = - k by INT_1:def 1;
A2:  k in NAT by ORDINAL1:def 12;
    per cases by A1;
      suppose
     x = k;
        hence thesis by XBOOLE_0:def 3,A2;
      end;
      suppose
     x = -k;
then      x in INT- by Def1,A2;
        hence thesis by XBOOLE_0:def 3;
      end;
  end;
then A3: INT c= INT- \/ NAT;
 for x being object st x in INT- \/ NAT holds x in INT
  proof
    let x be object;
    assume
A4: x in INT- \/ NAT;
 now per cases by A4,XBOOLE_0:def 3;
      suppose
    x in INT-;
then     ex k being Element of NAT st x = -k by Def1;
        hence thesis by INT_1:def 1;
      end;
      suppose
    x in NAT;
        hence thesis by INT_1:def 1;
      end;
    end;
    hence thesis;
  end;
then  INT- \/ NAT c= INT;
  hence thesis by A3,XBOOLE_0:def 10;
end;
