reserve T for TopSpace,
  A, B for Subset of T;

theorem Th27:
  for A being Subset of R^1, a, b being Real st A = ].
  -infty,a.] \/ IRRAT (a,b) \/ [.b,+infty .[ holds Cl A = the carrier of R^1
proof
  reconsider B = IRRAT as Subset of R^1 by TOPMETR:17;
  let A be Subset of R^1, a, b be Real;
  assume
A1: A = ]. -infty,a.] \/ IRRAT (a,b) \/ [.b,+infty .[;
  B c= A
  proof
    let x be object;
    assume
A2: x in B;
    then reconsider x9 = x as Real;
    per cases;
    suppose
      x9 <= a;
      then x9 in ]. -infty,a.] by XXREAL_1:234;
      then x9 in ]. -infty,a.] \/ IRRAT (a,b) by XBOOLE_0:def 3;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
    suppose
      x9 > a & x9 < b;
      then x9 in ]. a, b .[ by XXREAL_1:4;
      then x9 in IRRAT /\ ]. a, b .[ by A2,XBOOLE_0:def 4;
      then x9 in IRRAT (a, b) by BORSUK_5:def 3;
      then x9 in ]. -infty,a.] \/ IRRAT (a,b) by XBOOLE_0:def 3;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
    suppose
      x9 >= b;
      then x9 in [.b,+infty .[ by XXREAL_1:236;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
  end;
  then
A3: Cl B c= Cl A by PRE_TOPC:19;
  Cl B = the carrier of R^1 by BORSUK_5:28;
  hence thesis by A3,XBOOLE_0:def 10;
end;
