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

theorem
  UAp UAp X = UAp X
proof
  hereby
    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 y being object such that
A2: y in Class (the InternalRel of A, x) and
A3: y in UAp X by XBOOLE_0:3;
A4: Class (the InternalRel of A, y) = Class (the InternalRel of A, x)
      by A1,A2,EQREL_1:23;
    Class (the InternalRel of A, y) meets X by A3,Th10;
    hence x in UAp X by A1,A4;
  end;
  thus thesis by Th13;
end;
