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

theorem
  LAp (X /\ Y) = LAp X /\ LAp Y
proof
  thus LAp (X /\ Y) c= LAp X /\ LAp Y
  proof
    let x be object;
    assume
A1: x in LAp (X /\ Y);
    then Class (the InternalRel of A, x) c= Y by Th8,XBOOLE_1:18;
    then
A2: x in LAp Y by A1;
    Class (the InternalRel of A, x) c= X by A1,Th8,XBOOLE_1:18;
    then x in LAp X by A1;
    hence thesis by A2,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A3: x in LAp X /\ LAp Y;
  then x in LAp Y by XBOOLE_0:def 4;
  then
A4: Class (the InternalRel of A, x) c= Y by Th8;
  x in LAp X by A3,XBOOLE_0:def 4;
  then Class (the InternalRel of A, x) c= X by Th8;
  then Class (the InternalRel of A, x) c= X /\ Y by A4,XBOOLE_1:19;
  hence thesis by A3;
end;
