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_1:6
  for Z, M be ManySortedSet of I for SF be non-empty MSSubsetFamily of M
st (for Z1 be ManySortedSet of I st Z1 in SF holds Z c= Z1) holds Z c= meet SF
proof
  let Z, M be ManySortedSet of I, SF be non-empty MSSubsetFamily of M such
  that
A1: for Z1 be ManySortedSet of I st Z1 in SF holds Z c= Z1;
  let i be object;
  consider T be ManySortedSet of I such that
A2: T in SF by PBOOLE:134;
  assume
A3: i in I;
  then consider Q be Subset-Family of (M.i) such that
A4: Q = SF.i and
A5: (meet SF).i = Intersect Q by Def1;
A6: for Z9 be set st Z9 in Q holds Z.i c= Z9
  proof
    let Z9 be set such that
A7: Z9 in Q;
    dom (T +* (i .--> Z9)) = I by A3,PZFMISC1:1;
    then reconsider K = T +* (i .--> Z9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A8: dom (i .--> Z9) = {i};
    i in {i} by TARSKI:def 1;
    then
A9: K.i = (i .--> Z9).i by A8,FUNCT_4:13
      .= Z9 by FUNCOP_1:72;
    K in SF
    proof
      let q be object such that
A10:  q in I;
      per cases;
      suppose
        q = i;
        hence thesis by A4,A7,A9;
      end;
      suppose
        q <> i;
        then not q in dom (i .--> Z9) by TARSKI:def 1;
        then K.q = T.q by FUNCT_4:11;
        hence thesis by A2,A10;
      end;
    end;
    then Z c= K by A1;
    hence thesis by A3,A9;
  end;
  Intersect Q = meet Q by A3,A4,SETFAM_1:def 9;
  hence thesis by A3,A4,A5,A6,SETFAM_1:5;
end;
