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 Th41:
  SF = EmptyMS I implies meet SF = M
proof
  assume
A1: SF = EmptyMS I;
  now
    let i be object;
    assume i in I;
    then consider Q be Subset-Family of (M.i) such that
A2: Q = SF.i and
A3: (meet SF).i = Intersect Q by Def1;
    Q = {} by A1,A2;
    hence (meet SF).i = M.i by A3,SETFAM_1:def 9;
  end;
  hence thesis;
end;
