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_1:27
  (A c= rngs F & for i be set for f be Function st i in I & f = F.i
  holds f"(A.i) is finite) implies A is finite-yielding
proof
  assume that
A1: A c= rngs F and
A2: for i be set for f be Function st i in I & f = F.i holds f"(A.i) is finite;
  let i such that
A3: i in I;
  reconsider f = F.i as Function;
  (rngs F).i = rng f by A3,Th13;
  then
A4: A.i c= rng f by A1,A3;
  f"(A.i) is finite by A2,A3;
  hence thesis by A4,FINSET_1:9;
end;
