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

theorem
  UAp LAp UAp X = UAp X
proof
  thus UAp LAp UAp X c= UAp X
  proof
    let x be object;
    assume x in UAp LAp UAp X;
    then Class (the InternalRel of A, x) meets LAp UAp X by Th10;
    then consider y being object such that
A1: y in Class (the InternalRel of A, x) and
A2: y in LAp UAp X by XBOOLE_0:3;
    [y, x] in the InternalRel of A by A1,EQREL_1:19;
    then [x, y] in the InternalRel of A by EQREL_1:6;
    then
A3: x in Class (the InternalRel of A, y) by EQREL_1:19;
    Class (the InternalRel of A, y) c= UAp X by A2,Th8;
    hence thesis by A3;
  end;
  X c= LAp UAp X
  proof
    let x be object;
    assume
A4: x in X;
    Class (the InternalRel of A, x) c= UAp X
    proof
      let y be object;
      assume
A5:   y in Class (the InternalRel of A, x);
      then [y,x] in the InternalRel of A by EQREL_1:19;
      then [x,y] in the InternalRel of A by EQREL_1:6;
      then x in Class (the InternalRel of A, y) by EQREL_1:19;
      then Class (the InternalRel of A, y) meets X by A4,XBOOLE_0:3;
      hence thesis by A5;
    end;
    hence thesis by A4;
  end;
  hence UAp X c= UAp LAp UAp X by Th25;
end;
