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

theorem Th34:
  LAp LAp X = UAp LAp X
proof
  thus LAp LAp X c= UAp LAp X by Th14;
  let x be object;
  assume
A1: x in UAp LAp X;
  then Class (the InternalRel of A, x) meets LAp X by Th10;
  then consider z being object such that
A2: z in Class (the InternalRel of A, x) and
A3: z in LAp X by XBOOLE_0:3;
  Class (the InternalRel of A, z) c= X by A3,Th8;
  then
A4: Class (the InternalRel of A, x) c= X by A1,A2,EQREL_1:23;
  Class (the InternalRel of A, x) c= LAp X
  proof
    let y be object;
    assume
A5: y in Class (the InternalRel of A, x);
    then Class (the InternalRel of A, x) = Class (the InternalRel of A, y) by
A1,EQREL_1:23;
    hence thesis by A4,A5;
  end;
  hence thesis by A1;
end;
