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

theorem
  UAp (X \/ Y) = UAp X \/ UAp Y
proof
  thus UAp (X \/ Y) c= UAp X \/ UAp Y
  proof
    let x be object;
    assume
A1: x in UAp (X \/ Y);
    then Class (the InternalRel of A, x) meets (X \/ Y) by Th10; then
    Class (the InternalRel of A, x) meets X or
      Class (the InternalRel of A, x) meets Y by XBOOLE_1:70;
    then x in UAp X or x in UAp Y by A1;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume
A2: x in UAp X \/ UAp Y;
  then x in UAp X or x in UAp Y by XBOOLE_0:def 3;
  then
  Class (the InternalRel of A, x) meets X or Class (the InternalRel of A,
  x) meets Y by Th10;
  then Class (the InternalRel of A, x) meets (X \/ Y) by XBOOLE_1:70;
  hence thesis by A2;
end;
