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 Th21:
  A is finite-yielding iff bool A is finite-yielding
proof
  thus A is finite-yielding implies bool A is finite-yielding
  proof
    assume
A1: A is finite-yielding;
    let i be object;
    assume
A2: i in I;
    A.i is finite by A1;
    then bool (A.i) is finite;
    hence thesis by A2,MBOOLEAN:def 1;
  end;
  assume
A3: bool A is finite-yielding;
  let i be object;
  assume
A4: i in I;
  (bool A).i is finite by A3;
  then bool (A.i) is finite by A4,MBOOLEAN:def 1;
  hence thesis;
end;
