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 Th32:
  i in I implies SF.i is Subset-Family of (M.i)
proof
  assume
A1: i in I;
  SF c= bool M by PBOOLE:def 18;
  then SF.i c= (bool M).i by A1;
  hence thesis by A1,MBOOLEAN:def 1;
end;
