reserve A for Tolerance_Space,
  X, Y for Subset of A;

theorem
  BndAp X = BndAp X`
proof
  thus BndAp X c= BndAp X`
  proof
    let x be object;
    assume
A1: x in BndAp X;
    then x in UAp X by XBOOLE_0:def 5;
    then not x in (UAp X)` by XBOOLE_0:def 5;
    then
A2: not x in LAp X` by Th28;
    not x in LAp X by A1,XBOOLE_0:def 5;
    then x in (LAp X)` by A1,XBOOLE_0:def 5;
    then x in UAp X` by Th29;
    hence thesis by A2,XBOOLE_0:def 5;
  end;
  thus BndAp X` c= BndAp X
  proof
    let x be object;
    assume
A3: x in BndAp X`;
    then x in UAp X` by XBOOLE_0:def 5;
    then x in (LAp X)` by Th29;
    then
A4: not x in LAp X by XBOOLE_0:def 5;
    not x in LAp X` by A3,XBOOLE_0:def 5;
    then not x in (UAp X)` by Th28;
    then x in (UAp X)`` by A3,SUBSET_1:29;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
end;
