reserve a, I for set,
  S for non empty non void ManySortedSign;

theorem
  for A being ManySortedSet of I ex B being non-empty ManySortedSet of I
  st A c= B
proof
  let A be ManySortedSet of I;
  deffunc F(object) = {{}} \/ A.$1;
  consider f being ManySortedSet of I such that
A1: for i be object st i in I holds f.i = F(i) from PBOOLE:sch 4;
  f is non-empty
  proof
    let i be object;
    assume i in I;
    then f.i = {{}} \/ A.i by A1;
    hence thesis;
  end;
  then reconsider f as non-empty ManySortedSet of I;
  take f;
  let i be object;
  assume i in I;
  then f.i = A.i \/ {{}} by A1;
  hence thesis by XBOOLE_1:7;
end;
