theorem Th57:
  for FSi being FinSequence of Si holds Union FSi in Si
proof
  let FSi be FinSequence of Si;
  consider ASeq being SetSequence of Si such that
A1: ( for k st k in dom FSi holds ASeq.k = FSi.k)&
    for k st not k in dom FSi holds ASeq.k = {} by Th56;
  reconsider ASeq as SetSequence of X;
 ( for k st k in dom FSi holds ASeq.k = FSi.k)&
    for k st not k in dom FSi holds ASeq.k = {} by A1;
  then Union ASeq = Union FSi by Th55;
  hence thesis by PROB_1:17;
end;
