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
  union A is finite-yielding implies A is finite-yielding & for M st M
  in A holds M is finite-yielding
proof
  assume
A1: union A is finite-yielding;
  thus A is finite-yielding
  proof
    let i be object;
    assume
A2: i in I;
    (union A).i is finite by A1;
    then union (A.i) is finite by A2,MBOOLEAN:def 2;
    hence thesis by FINSET_1:7;
  end;
  let M such that
A3: M in A;
  let i be object;
  assume
A4: i in I;
  (union A).i is finite by A1;
  then
A5: union (A.i) is finite by A4,MBOOLEAN:def 2;
  M.i in A.i by A3,A4;
  hence thesis by A5,FINSET_1:7;
end;
