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_1:8
  A in SF & A c= B implies meet SF c= B
proof
  assume that
A1: A in SF and
A2: A c= B;
  let i be object;
  assume
A3: i in I;
  then
A4: A.i c= B.i by A2;
  consider Q be Subset-Family of (M.i) such that
A5: Q = SF.i and
A6: (meet SF).i = Intersect Q by A3,Def1;
A7: A.i in SF.i by A1,A3;
  then Intersect Q = meet Q by A5,SETFAM_1:def 9;
  hence thesis by A5,A6,A7,A4,SETFAM_1:7;
end;
