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 Th56:
  for FSi being FinSequence of Si holds ex ASeq being SetSequence
  of Si st (for k st k in dom FSi holds ASeq.k = FSi.k) & for k st not k in dom
  FSi holds ASeq.k = {}
proof
  let FSi be FinSequence of Si;
  consider A1 being SetSequence of X such that
A1: for k st k in dom FSi holds A1.k = FSi.k and
A2: for k st not k in dom FSi holds A1.k = {} by Th54;
  for n being Nat holds A1.n in Si
  proof
    let n be Nat;
    per cases;
    suppose
      not n in dom FSi;
      then A1.n = {} by A2;
      hence thesis by PROB_1:4;
    end;
    suppose
      n in dom FSi;
      then A1.n = FSi.n by A1;
      hence thesis;
    end;
  end;
  then rng A1 c= Si by NAT_1:52;
  then reconsider A1 as SetSequence of Si by RELAT_1:def 19;
  take A1;
  thus thesis by A1,A2;
end;
