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

theorem Th16:
  X is exact iff UAp X = X
proof
  hereby
    assume X is exact;
    then BndAp X = {};
    then
A1: UAp X c= LAp X by XBOOLE_1:37;
A2: X c= UAp X by Th13;
    LAp X c= X by Th12;
    then UAp X c= X by A1;
    hence UAp X = X by A2;
  end;
  assume
A3: UAp X = X;
  X c= LAp X
  proof
    let x be object;
    assume
A4: x in X;
    Class (the InternalRel of A, x) c= X
    proof
      let y be object;
      assume
A5:   y in Class (the InternalRel of A, x);
      then [y,x] in the InternalRel of A by EQREL_1:19;
      then [x,y] in the InternalRel of A by EQREL_1:6;
      then x in Class (the InternalRel of A, y) by EQREL_1:19;
      then Class (the InternalRel of A, y) meets X by A4,XBOOLE_0:3;
      hence thesis by A3,A5;
    end;
    hence thesis by A4;
  end;
  then BndAp X = {} by A3,XBOOLE_1:37;
  hence thesis;
end;
