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

theorem Th36:
  UAp UAp X = LAp UAp X
proof
  thus UAp UAp X c= LAp UAp X
  proof
    let x be object;
    assume
A1: x in UAp UAp X;
    then Class (the InternalRel of A, x) meets UAp X by Th10;
    then consider z being object such that
A2: z in Class (the InternalRel of A, x) and
A3: z in UAp X by XBOOLE_0:3;
A4: Class (the InternalRel of A, z) = Class (the InternalRel of A, x)
      by A1,A2,EQREL_1:23;
A5: Class (the InternalRel of A, z) meets X by A3,Th10;
    Class (the InternalRel of A, x) c= UAp X
    proof
      let y be object;
      assume
A6:   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 A5,A4,A6;
    end;
    hence thesis by A1;
  end;
  thus thesis by Th14;
end;
