reserve r, s, t for Real;
reserve seq for Real_Sequence,
  X, Y for Subset of REAL;
reserve r3, r1, q3, p3 for Real;

theorem
  for X being without_zero Subset of REAL st X is closed real-bounded
  holds Inv X is closed
proof
  let X be without_zero Subset of REAL;
  assume that
A1: X is closed and
A2: X is real-bounded;
  let s1 be Real_Sequence;
  assume that
A3: (rng s1) c= Inv X and
A4: s1 is convergent;
A5: rng (s1") c= X
  proof
    let x be object;
    assume x in rng (s1");
    then consider n being object such that
A6: n in dom (s1") and
A7: x = (s1").n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A6;
    s1.n in rng s1 by FUNCT_2:4;
    then 1/(s1.n) in Inv Inv X by A3;
    hence thesis by A7,VALUED_1:10;
  end;
A8: not 0 in rng s1 by A3;
A9: now assume not s1 is non-zero;
    then consider n being Nat such that
A10:   s1.n = 0 by SEQ_1:5;
   n in NAT by ORDINAL1:def 12;
    hence contradiction by A8,FUNCT_2:4,A10;
  end;
A11: now
A12: rng (s1") c= X
    proof
      let y be object;
      assume y in rng (s1");
      then consider x being object such that
A13:  x in dom (s1") and
A14:  y = (s1").x by FUNCT_1:def 3;
      reconsider x as Element of NAT by A13;
      s1.x in rng s1 by FUNCT_2:4;
      then 1/(s1.x) in Inv Inv X by A3;
      hence thesis by A14,VALUED_1:10;
    end;
    assume lim s1 = 0;
    then s1" is non bounded by A4,A9,Th38;
    then rng (s1") is non real-bounded by Th39;
    hence contradiction by A2,A12,XXREAL_2:45;
  end;
  then s1" is convergent by A4,A9,SEQ_2:21;
  then lim s1" in X by A1,A5;
  then 1/lim s1 in X by A4,A9,A11,SEQ_2:22;
  then 1/(1/lim s1) in Inv X;
  hence thesis;
end;
