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

theorem Th32:
  for A being Subset of R^1, a, b, c being Real st A = ].
  -infty, a .[ \/ RAT (b,c) & a < b & b < c holds Int A = ]. -infty, a .[
proof
  let A be Subset of R^1, a, b, c be Real;
  reconsider B = [. a, b .], C = IRRAT (b,c), D = [. c,+infty .[ as Subset of
  R^1 by TOPMETR:17;
  assume that
A1: A = ]. -infty, a.[ \/ RAT (b,c) and
A2: a < b and
A3: b < c;
A4: a < c by A2,A3,XXREAL_0:2;
  A` = REAL \ ( ]. -infty, a.[ \/ RAT (b,c)) by A1,TOPMETR:17
    .= (REAL \ RAT (b,c)) \ ]. -infty, a.[ by XBOOLE_1:41
    .= (]. -infty,b.] \/ IRRAT (b, c) \/ [.c,+infty .[) \ ]. -infty,a.[ by
BORSUK_5:58
    .= (]. -infty,b.] \/ (IRRAT (b, c) \/ [.c,+infty .[)) \ ]. -infty,a.[ by
XBOOLE_1:4
    .= (]. -infty,b.] \ ]. -infty,a.[) \/ ((IRRAT (b, c) \/ [.c,+infty .[) \
  ]. -infty,a.[) by XBOOLE_1:42;
  then
A5: A` = [.a,b.] \/ ((IRRAT (b, c) \/ [.c,+infty .[) \ ]. -infty,a.[) by
XXREAL_1:289
    .= [.a,b.] \/ ((IRRAT (b,c) \ ]. -infty,a.[) \/ ([.c,+infty .[ \ ].
  -infty,a.[)) by XBOOLE_1:42;
  right_closed_halfline c is closed;
  then D is closed by JORDAN5A:23;
  then
A6: Cl D = D by PRE_TOPC:22;
  [.b,+infty .[ misses ]. -infty,a.[ by A2,XXREAL_1:94;
  then IRRAT (b,c) misses ]. -infty,a.[ by Th31,XBOOLE_1:63;
  then
A7: IRRAT (b,c) \ ]. -infty,a.[ = IRRAT (b,c) by XBOOLE_1:83;
  B is closed by JORDAN5A:23;
  then
A8: Cl B = B by PRE_TOPC:22;
  [.c,+infty .[ misses ]. -infty,a.[ by A2,A3,XXREAL_0:2,XXREAL_1:94;
  then A` = [.a,b.] \/ (IRRAT (b,c) \/ [.c,+infty .[) by A5,A7,XBOOLE_1:83
    .= [.a,b.] \/ (IRRAT (b,c) \/ ({c} \/ ].c,+infty .[)) by BORSUK_5:43
    .= [.a,b.] \/ IRRAT (b,c) \/ ({c} \/ ].c,+infty .[) by XBOOLE_1:4
    .= [. a, b .] \/ IRRAT (b,c) \/ [. c,+infty .[ by BORSUK_5:43;
  then Cl A` = Cl (B \/ C) \/ Cl D by PRE_TOPC:20
    .= Cl B \/ Cl C \/ Cl D by PRE_TOPC:20
    .= B \/ [.b,c.] \/ D by A8,A6,A3,BORSUK_5:32
    .= [.a,c.] \/ D by A2,A3,XXREAL_1:165
    .= [. a,+infty .[ by A4,BORSUK_5:11;
  then (Cl A`)` = ]. -infty, a .[ by TOPMETR:17,XXREAL_1:224,294;
  hence thesis by TOPS_1:def 1;
end;
