reserve W for Universe,
  H for ZF-formula,
  x,y,z,X for set,
  k for Variable,
  f for Function of VAR,W,
  u,v for Element of W;
reserve F for Function,
  A,B,C for Ordinal,
  a,b,b1,b2,c for Ordinal of W,
  fi for Ordinal-Sequence,
  phi for Ordinal-Sequence of W,
  H for ZF-formula;
reserve psi for Ordinal-Sequence;
reserve L for DOMAIN-Sequence of W,
  n for Element of NAT,
  f for Function of VAR,L.a;

theorem Th17:
  NAT,VAR are_equipotent
proof
  deffunc F(Nat,set) = 5+($1+1);
  consider f being sequence of NAT such that
A1: f.0 = 5+0 & for n being Nat holds f.(n+1) = F(n,f.n) from NAT_1:sch
  12;
A2: now
    let n;
    (ex j be Nat st n = j+1) implies f.n = 5+n by A1;
    then n = 0 or f.n = 5+n by NAT_1:6;
    hence f.n = 5+n by A1;
  end;
A3: dom f = NAT by FUNCT_2:def 1;
  thus NAT,VAR are_equipotent
  proof
    reconsider g = f as Function;
    take g;
    thus g is one-to-one
    proof
      let x,y be object;
      assume x in dom g & y in dom g;
      then reconsider n1 = x, n2 = y as Element of NAT by FUNCT_2:def 1;
      f.n1 = 5+n1 & f.n2 = 5+n2 by A2;
      hence thesis by XCMPLX_1:2;
    end;
    thus dom g = NAT by FUNCT_2:def 1;
    thus rng g c= VAR
    proof
      let x be object;
      assume x in rng g;
      then consider y being object such that
A4:   y in dom f and
A5:   x = f.y by FUNCT_1:def 3;
      reconsider y as Element of NAT by A4;
A6:   5+y >= 5 by NAT_1:11;
      x = 5+y by A2,A5;
      hence thesis by A6;
    end;
    let x be object;
    assume x in VAR;
    then ex n st x = n & 5 <= n;
    then consider n be Nat such that
A7: x = 5+n by NAT_1:10;
A8: n in NAT by ORDINAL1:def 12;
    then f.n = x by A2,A7;
    hence thesis by A3,A8,FUNCT_1:def 3;
  end;
end;
