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_1:4
  A in SF implies meet SF c= A
proof
  assume
A1: A in SF;
  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;
A5: A.i in SF.i by A1,A2;
  then Intersect Q = meet Q by A3,SETFAM_1:def 9;
  hence thesis by A3,A4,A5,SETFAM_1:3;
end;
