reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;

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;
