reserve a, b, r, s for Real;

theorem Th32:
  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 MEASURE6:59;
A4: [.a,b.] /\ (left_closed_halfline(a) \/ right_closed_halfline(b)) = {a,b
  } by A1,Th8;
  set LO = R^1(left_open_halfline(a)), RC = R^1(right_closed_halfline(b)), RO
  = R^1(right_open_halfline(b)), LC = R^1(left_closed_halfline(a));
A5: RC = right_closed_halfline(b) by TOPREALB:def 3;
A6: LC = left_closed_halfline(a) by TOPREALB:def 3;
A7: RO = right_open_halfline(b) by TOPREALB:def 3;
A8: LO = left_open_halfline(a) by TOPREALB:def 3;
  then
A9: [.a,b.]` = LO \/ RO by A7,XXREAL_1:385;
  Cl X` = Cl [.a,b.]` by A2,JORDAN5A:24,TOPMETR:17
    .= Cl left_open_halfline(a) \/ Cl right_open_halfline(b) by A8,A7,A9,Th3
    .= Cl LO \/ Cl right_open_halfline(b) by A8,JORDAN5A:24
    .= Cl LO \/ Cl RO by A7,JORDAN5A:24
    .= LC \/ Cl RO by A6,BORSUK_5:51,TOPREALB:def 3
    .= LC \/ RC by A5,BORSUK_5:49,TOPREALB:def 3
    .= left_closed_halfline(a) \/ right_closed_halfline(b) by A5,TOPREALB:def 3
;
  hence thesis by A3,A4,TOPS_1:def 2;
end;
