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
  A in meet SF implies for B st B in SF holds A in B
proof
  assume
A1: A in meet SF;
  let B such that
A2: B in SF;
  let i be object;
  assume
A3: i in I;
  then
A4: A.i in (meet SF).i by A1;
A5: B.i in SF.i by A2,A3;
  ex Q be Subset-Family of (M.i) st Q = SF.i & (meet SF).i = Intersect Q by A3
,Def1;
  hence thesis by A4,A5,SETFAM_1:43;
end;
