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

theorem
  LAp LAp X = LAp X
proof
  thus LAp LAp X c= LAp X by Th12;
  let x be object;
  assume
A1: x in LAp X;
  then
A2: Class (the InternalRel of A, x) c= X by Th8;
  Class (the InternalRel of A, x) c= LAp X
  proof
    let y be object;
    assume
A3: 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 A2,A3;
  end;
  hence thesis by A1;
end;
