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

theorem
  LAp X \/ LAp Y c= LAp (X \/ Y)
proof
  let x be object;
  assume
A1: x in LAp X \/ LAp Y;
  per cases by A1,XBOOLE_0:def 3;
  suppose
    x in LAp X;
    then Class (the InternalRel of A, x) c= X \/ Y by Th8,XBOOLE_1:10;
    hence thesis by A1;
  end;
  suppose
    x in LAp Y;
    then Class (the InternalRel of A, x) c= X \/ Y by Th8,XBOOLE_1:10;
    hence thesis by A1;
  end;
end;
