reserve a, b, r, s for Real;

theorem
  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 = Cl ].a,b.[ by A2,JORDAN5A:24
    .= [.a,b.] by A1,JORDAN5A:26;
  set RC = R^1(right_closed_halfline(b)), LC = R^1(left_closed_halfline(a));
A4: RC = right_closed_halfline(b) & LC = left_closed_halfline(a) by
TOPREALB:def 3;
  then
A5: ].a,b.[` = LC \/ RC by XXREAL_1:398;
A6: [.a,b.] /\ (left_closed_halfline(a) \/ right_closed_halfline(b)) = {a,b}
  by A1,Th8;
  Cl X` = Cl ].a,b.[` by A2,JORDAN5A:24,TOPMETR:17
    .= Cl left_closed_halfline(a) \/ Cl right_closed_halfline(b) by A4,A5,Th3
    .= Cl left_closed_halfline(a) \/ right_closed_halfline(b) by MEASURE6:59
    .= left_closed_halfline(a) \/ right_closed_halfline(b) by MEASURE6:59;
  hence thesis by A3,A6,TOPS_1:def 2;
end;
