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 :: SETFAM_1:5
  EmptyMS I in SF implies meet SF = EmptyMS I
proof
  assume
A1: EmptyMS I in SF;
  now
    let i be object;
    assume
A2: i in I;
    then consider Q be Subset-Family of (M.i) such that
A3: Q = SF.i and
A4: (meet SF).i = Intersect Q by Def1;
    EmptyMS I.i in Q by A1,A2,A3;
    then
A5: {} in Q;
    EmptyMS I.i in SF.i by A1,A2;
    then Intersect Q = meet Q by A3,SETFAM_1:def 9;
    then Intersect Q = {} by A5,SETFAM_1:4;
    hence (meet SF).i = EmptyMS I.i by A4;
  end;
  hence thesis;
end;
