theorem
  for TS being non empty set, A, B being with_proper_subsets
  Subset-Family of TS holds A \/ B is with_proper_subsets
proof
  let TS be non empty set, A, B be with_proper_subsets Subset-Family of TS;
  assume TS in A \/ B;
  then TS in A or TS in B by XBOOLE_0:def 3;
  hence thesis by Def12;
end;
