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 B be non-empty ManySortedSet of I holds A is Element of B implies A in B
proof
  let B be non-empty ManySortedSet of I;
  assume
A1: A is Element of B;
  let i be object;
  assume
A2: i in I;
  then
A3: B.i <> {};
  A.i is Element of B.i by A1,A2,PBOOLE:def 14;
  hence thesis by A3;
end;
