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
  for A, M be ManySortedSet of I for SF be non-empty MSSubsetFamily of M
  st (A in M & for B be ManySortedSet of I st B in SF holds A in B) holds A in
  meet SF
proof
  let A, M be ManySortedSet of I, SF be non-empty MSSubsetFamily of M;
  assume that
A1: A in M and
A2: for B be ManySortedSet of I st B in SF holds A in B;
  let i be object;
  consider T be ManySortedSet of I such that
A3: T in SF by PBOOLE:134;
  assume
A4: i in I;
  then consider Q be Subset-Family of (M.i) such that
A5: Q = SF.i and
A6: (meet SF).i = Intersect Q by Def1;
A7: for B9 be set st B9 in Q holds A.i in B9
  proof
    let B9 be set such that
A8: B9 in Q;
    dom (T +* (i .--> B9)) = I by A4,PZFMISC1:1;
    then reconsider K = T +* (i .--> B9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A9: dom (i .--> B9) = {i};
    i in {i} by TARSKI:def 1;
    then
A10: K.i = (i .--> B9).i by A9,FUNCT_4:13
      .= B9 by FUNCOP_1:72;
    K in SF
    proof
      let q be object such that
A11:  q in I;
      per cases;
      suppose
        q = i;
        hence thesis by A5,A8,A10;
      end;
      suppose
        q <> i;
        then not q in dom (i .--> B9) by TARSKI:def 1;
        then K.q = T.q by FUNCT_4:11;
        hence thesis by A3,A11;
      end;
    end;
    then A in K by A2;
    hence thesis by A4,A10;
  end;
  A.i in M.i by A1,A4;
  hence thesis by A6,A7,SETFAM_1:43;
end;
