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;

theorem
  for A be non-empty ManySortedSet of I holds A is finite-yielding & (
for M be ManySortedSet of I st M in A holds M is finite-yielding) implies union
  A is finite-yielding
proof
  let A be non-empty ManySortedSet of I;
  assume that
A1: A is finite-yielding and
A2: for M be ManySortedSet of I st M in A holds M is finite-yielding;
  let i be object;
  assume
A3: i in I;
A4: for X9 be set st X9 in A.i holds X9 is finite
  proof
    consider M be ManySortedSet of I such that
A5: M in A by PBOOLE:134;
    let X9 be set such that
A6: X9 in A.i;
    dom (M +* (i .--> X9)) = I by A3,PZFMISC1:1;
    then reconsider K = M +* (i .--> X9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A7: dom (i .--> X9) = {i};
    i in {i} by TARSKI:def 1;
    then
A8: K.i = (i .--> X9).i by A7,FUNCT_4:13
      .= X9 by FUNCOP_1:72;
    K in A
    proof
      let j be object such that
A9:   j in I;
      now
        per cases;
        case
          j = i;
          hence thesis by A6,A8;
        end;
        case
          j <> i;
          then not j in dom (i .--> X9) by TARSKI:def 1;
          then K.j = M.j by FUNCT_4:11;
          hence thesis by A5,A9;
        end;
      end;
      hence thesis;
    end;
    then K is finite-yielding by A2;
    hence thesis by A8;
  end;
  A.i is finite by A1;
  then union (A.i) is finite by A4,FINSET_1:7;
  hence thesis by A3,MBOOLEAN:def 2;
end;
