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

theorem Th15:
  X is exact iff LAp X = X
proof
A1: LAp X c= UAp X by Th14;
  hereby
    assume X is exact;
    then BndAp X = {};
    then UAp X c= LAp X by XBOOLE_1:37;
    then UAp X = LAp X by A1;
    then
A2: X c= LAp X by Th13;
    LAp X c= X by Th12;
    hence LAp X = X by A2;
  end;
  assume
A3: LAp X = X;
  UAp X c= X
  proof
    let y be object;
    assume y in UAp X;
    then Class (the InternalRel of A, y) meets X by Th10;
    then consider z being object such that
A4: z in Class (the InternalRel of A, y) & z in LAp X by A3,XBOOLE_0:3;
    Class (the InternalRel of A, z) c= X & y in Class (the InternalRel of
    A, z) by A4,Th7,Th8;
    hence thesis;
  end;
  then BndAp X = {} by A3,XBOOLE_1:37;
  hence thesis;
end;
