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

theorem Th1:
  NAT,INT- are_equipotent
proof
  defpred P[Element of NAT,set] means $2=-$1;
A1: for x being Element of NAT ex y being Element of INT- st P[x,y]
  proof
    let x be Element of NAT;
 -x in INT- by Def1;
    hence thesis;
  end;
  consider f being sequence of INT- such that
A2: for x being Element of NAT holds P[x,f.x] from FUNCT_2:sch 3(A1);
A3: f in Funcs(NAT,INT-) by FUNCT_2:8;
then A4: dom f = NAT by FUNCT_2:92;
A5: for x1,x2 being object st x1 in dom f & x2 in dom f & f.x1=f.x2
holds x1=x2
  proof
    let x1,x2 be object;
    assume that
A6: x1 in dom f and
A7: x2 in dom f and
A8: f.x1=f.x2;
    reconsider x1 as Element of NAT by A3,A6,FUNCT_2:92;
    reconsider x2 as Element of NAT by A3,A7,FUNCT_2:92;
 f.x1 = -x1 & f.x2 = -x2 by A2;
then  --x1=x2 by A8;
    hence thesis;
  end;
A9: for y being object st y in INT- holds f"{y} <> {}
  proof
    let y being object;
    assume
A10: y in INT-;
    then reconsider y as Real;
    consider k being Element of NAT such that
A11: y = -k by A10,Def1;
 f.k = -k by A2;
then  f.k in {y} by A11,TARSKI:def 1;
    hence thesis by A4,FUNCT_1:def 7;
  end;
A12: f is one-to-one by A5,FUNCT_1:def 4;
 rng f = INT- by A9,FUNCT_2:41;
  hence thesis by A4,A12,WELLORD2:def 4;
end;
