reserve a, I for set,
  S for non empty non void ManySortedSign;

theorem
  for M being non-empty ManySortedSet of I for B being finite-yielding
ManySortedSubset of M ex C being non-empty finite-yielding ManySortedSubset of
  M st B c= C
proof
  let M be non-empty ManySortedSet of I, B be finite-yielding ManySortedSubset
  of M;
  defpred P[object,object] means
    ex a being Element of M.$1 st $2 = {a} \/ B.$1;
A1: now
    let i be object such that
    i in I;
    set a = the Element of M.i;
     reconsider j = {a} \/ B.i as object;
    take j;
    thus P[i,j];
  end;
  consider C being ManySortedSet of I such that
A2: for i be object st i in I holds P[i,C.i] from PBOOLE:sch 3(A1);
A3: C is ManySortedSubset of M
  proof
    let i be object;
    assume
A4: i in I;
    then consider a being Element of M.i such that
A5: C.i = {a} \/ B.i by A2;
    let q be object;
    assume q in C.i;
    then
A6: q in {a} or q in B.i by A5,XBOOLE_0:def 3;
    B c= M by PBOOLE:def 18;
    then B.i c= M.i by A4;
    then
A7: q = a or q in M.i by A6,TARSKI:def 1;
    M.i is non empty by A4;
    hence thesis by A7;
  end;
A8: C is finite-yielding
  proof
    let i be object;
    assume
A9: i in I;
    reconsider b = B.i as finite set;
    consider a being Element of M.i such that
A10: C.i = {a} \/ B.i by A2,A9;
    thus thesis by A10;
  end;
  C is non-empty
  proof
    let i be object;
    assume i in I;
    then ex a being Element of M.i st C.i = {a} \/ B.i by A2;
    hence thesis;
  end;
  then reconsider C as non-empty finite-yielding ManySortedSubset of M by A3,A8
;
  take C;
  let i be object;
  assume i in I;
  then ex a being Element of M.i st C.i = {a} \/ B.i by A2;
  hence thesis by XBOOLE_1:7;
end;
