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 :: FINSET_1:23
  A is non-empty & [|A,B|] is finite-yielding implies B is finite-yielding
proof
  assume that
A1: A is non-empty and
A2: [|A,B|] is finite-yielding;
  let i be object;
  assume
A3: i in I;
  [|A,B|].i is finite by A2;
  then [:A.i,B.i:] is finite by A3,PBOOLE:def 16;
  hence thesis by A1,A3;
end;
