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

theorem Th34:
  for A being Subset of R^1, a, b, c being Real st A = ].
-infty, a .] \/ [.b,c.] & a < b & b < c holds Int A = ]. -infty, a .[ \/ ].b,c
  .[
proof
  let A be Subset of R^1, a, b, c be Real;
  assume that
A1: A = ]. -infty, a .] \/ [.b,c.] and
A2: a < b and
A3: b < c;
  a < c by A2,A3,XXREAL_0:2;
  then
A4: ].c,+infty .[ /\ ].a,+infty .[ = ].c,+infty .[ by XBOOLE_1:28,XXREAL_1:46;
  reconsider B = ]. a,b .[, C = ].c,+infty .[ as Subset of R^1 by TOPMETR:17;
A5: Cl B = [. a,b .] by A2,BORSUK_5:16;
A6: Cl C = [. c,+infty .[ by BORSUK_5:49;
  A` = REAL \ (]. -infty, a .] \/ [.b,c.]) by A1,TOPMETR:17
    .= (REAL \ left_closed_halfline a) \ [.b,c.] by XBOOLE_1:41
    .= right_open_halfline a \ [.b,c.] by XXREAL_1:224,288
    .= ].a,+infty .[ \ ([.b,+infty .[ \ ].c,+infty .[) by XXREAL_1:295
    .= ( ].a,+infty .[ \ [.b,+infty .[) \/ ( ].a,+infty .[ /\ ].c,+infty .[)
  by XBOOLE_1:52
    .= ]. a,b .[ \/ ]. c,+infty .[ by A4,XXREAL_1:294;
  then (Cl A`)` = REAL \ ([. c,+infty .[ \/ [.a,b.]) by A5,A6,PRE_TOPC:20
,TOPMETR:17
    .= (REAL \ right_closed_halfline c) \ [.a,b.] by XBOOLE_1:41
    .= left_open_halfline c \ [.a,b.] by XXREAL_1:224,294
    .= ]. -infty, a .[ \/ ].b,c.[ by A2,A3,XXREAL_0:2,XXREAL_1:339;
  hence thesis by TOPS_1:def 1;
end;
