reserve a, b, r, s for Real;

theorem Th35:
  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_5:36;
A4: ].a,b.]` = left_closed_halfline(a) \/ right_open_halfline(b) by
XXREAL_1:399;
  set RO = R^1(right_open_halfline(b)), LC = R^1(left_closed_halfline(a));
A5: RO = right_open_halfline(b) by TOPREALB:def 3;
A6: LC = left_closed_halfline(a) by TOPREALB:def 3;
  Cl X` = Cl ].a,b.]` by A2,JORDAN5A:24,TOPMETR:17
    .= Cl left_closed_halfline(a) \/ Cl right_open_halfline(b) by A4,Th3
    .= Cl LC \/ Cl right_open_halfline(b) by A6,JORDAN5A:24
    .= Cl LC \/ Cl RO by A5,JORDAN5A:24
    .= LC \/ Cl RO by PRE_TOPC:22
    .= left_closed_halfline(a) \/ right_closed_halfline(b) by A6,BORSUK_5:49
,TOPREALB:def 3;
  hence thesis by A3,TOPS_1:def 2;
end;
