
theorem Th31:
  for A being Subset of REAL? st A = {REAL} holds A is closed
proof
  reconsider B = REAL \ NAT as Subset of R^1 by TOPMETR:17;
  let A be Subset of REAL?;
  reconsider B as Subset of R^1;
A1: (REAL\NAT) = ((REAL\NAT) \/ {REAL})\{REAL}
  proof
    thus (REAL\NAT) c= ((REAL\NAT) \/ {REAL})\{REAL}
    proof
      let x be object;
      assume
A2:   x in REAL \ NAT;
      then
A3:   x in REAL;
A4:   not x in {REAL}
      proof
        assume x in {REAL};
        then A: x = REAL by TARSKI:def 1;
        reconsider xx = x as set by TARSKI:1;
        not xx in xx; 
        hence contradiction by A,A3;
      end;
      x in (REAL \ NAT) \/ {REAL} by A2,XBOOLE_0:def 3;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    let x be object;
    assume
A5: x in ((REAL\NAT) \/ {REAL})\{REAL};
    then not x in {REAL} by XBOOLE_0:def 5;
    hence thesis by A5,XBOOLE_0:def 3;
  end;
  B misses NAT by XBOOLE_1:79;
  then
A6: B /\ NAT = {};
  then reconsider C=B as Subset of REAL? by Th29;
  assume A = {REAL};
  then
A7: C = A` by A1,Def8;
  B is open
  proof
    reconsider N=NAT as Subset of R^1 by TOPMETR:17,NUMBERS:19;
    reconsider N as Subset of R^1;
    N is closed & N` = B by Th10,TOPMETR:17;
    hence thesis by TOPS_1:3;
  end;
  then C is open by A6,Th30;
  hence thesis by A7,TOPS_1:3;
end;
