
theorem Th10:
  for B being Subset of R^1 holds B = NAT implies B is closed
proof
  let B be Subset of R^1;
A1: dom (id NAT) = NAT;
A2: rng (id NAT) = NAT;
  then reconsider S=(id NAT) as sequence of R^1
        by A1,FUNCT_2:2,TOPMETR:17,NUMBERS:19;
 for n being Element of NAT holds S.n in ].n-1/4,n + 1/4.[
  proof
    let n be Element of NAT;
    reconsider x=S.n as Real;
A3: - 1/4 + n < 0 + n by XREAL_1:8;
    x=n & n < n + 1/4 by XREAL_1:29;
    then x in {r where r is Real: n - 1/4 < r & r < n + 1/4} by A3;
    hence thesis by RCOMP_1:def 2;
  end;
  hence thesis by A2,Th9;
end;
