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

theorem
  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 Y by Th10,XBOOLE_1:74;
  then
A2: x in UAp Y by A1;
  Class (the InternalRel of A, x) meets X by A1,Th10,XBOOLE_1:74;
  then x in UAp X by A1;
  hence thesis by A2,XBOOLE_0:def 4;
end;
