reserve A for Tolerance_Space,
  X, Y for Subset of A;
reserve A for Approximation_Space,
  X for Subset of A;
reserve A for finite Tolerance_Space,
  X for Subset of A,
  x for Element of A;
reserve A for finite Approximation_Space,
  X, Y for Subset of A,
  x for Element of A;

theorem Th43:
  for A being discrete Approximation_Space, X being Subset of A
  holds X is exact
proof
  let A be discrete Approximation_Space, X be Subset of A;
A1: the InternalRel of A = id the carrier of A by ORDERS_3:def 1;
  X = UAp X
  proof
    thus X c= UAp X by Th13;
    let x be object;
    assume
A2: x in UAp X;
    then Class (the InternalRel of A, x) meets X by Th10;
    then
A3: ex y being object st y in Class (the InternalRel of A, x) & y in X
      by XBOOLE_0:3;
    Class (the InternalRel of A, x) = {x} by A1,A2,EQREL_1:25;
    hence thesis by A3,TARSKI:def 1;
  end;
  hence thesis;
end;
