
theorem
  for V being non empty addLoopStr, M,N being Subset of V st M is empty
  or N is empty holds M + N is empty
proof
  let V be non empty addLoopStr;
  let M,N be Subset of V;
  assume
A1: M is empty or N is empty;
  assume not M + N is empty;
  then consider x being object such that
A2: x in M + N;
  x in {u + v where u,v is Element of V: u in M & v in N} by A2,RUSUB_4:def 9;
  then ex u,v being Element of V st x = u + v & u in M & v in N;
  hence contradiction by A1;
end;
