reserve a, b, r, s for Real;

theorem Th34:
  for X being Subset of R^1 st a < b & X = [.a,b.[ holds Fr X = {a ,b}
proof
  let X be Subset of R^1 such that
A1: a < b and
A2: X = [.a,b.[;
A3: Cl X = [.a,b.] & [.a,b.] /\ (left_closed_halfline(a) \/
  right_closed_halfline(b)) = {a,b} by A1,A2,Th8,BORSUK_5:35;
  set LO = R^1(left_open_halfline(a)), RC = R^1(right_closed_halfline(b)), LC
  = R^1(left_closed_halfline(a));
A4: RC = right_closed_halfline(b) by TOPREALB:def 3;
A5: LO = left_open_halfline(a) by TOPREALB:def 3;
  then
A6: [.a,b.[` = LO \/ RC by A4,XXREAL_1:382;
A7: LC = left_closed_halfline(a) by TOPREALB:def 3;
  Cl X` = Cl [.a,b.[` by A2,JORDAN5A:24,TOPMETR:17
    .= Cl left_open_halfline(a) \/ Cl right_closed_halfline(b) by A5,A4,A6,Th3
    .= Cl LO \/ Cl right_closed_halfline(b) by A5,JORDAN5A:24
    .= Cl LO \/ Cl RC by A4,JORDAN5A:24
    .= LC \/ Cl RC by A7,BORSUK_5:51,TOPREALB:def 3
    .= left_closed_halfline(a) \/ right_closed_halfline(b) by A4,A7,PRE_TOPC:22
;
  hence thesis by A3,TOPS_1:def 2;
end;
