reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;
reserve X, Y, Z for ManySortedSet of I;
reserve SF, SG, SH for MSSubsetFamily of M,
  SFe for non-empty MSSubsetFamily of M,
  V, W for ManySortedSubset of M;

theorem Th34:
  SF (\/) SG is MSSubsetFamily of M
proof
  SF c= bool M & SG c= bool M by PBOOLE:def 18;
  then SF (\/) SG c= bool M by PBOOLE:16;
  hence thesis by PBOOLE:def 18;
end;
