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;
reserve FSeq for FinSequence of Sigma;

theorem
  P.(Union FSeq) <= Sum(P * FSeq) & (FSeq is disjoint_valued implies P.(
  Union FSeq) = Sum(P * FSeq))
proof
  consider ASeq being SetSequence of Sigma such that
A1: for k st k in dom FSeq holds ASeq.k = FSeq.k and
A2: for k st not k in dom FSeq holds ASeq.k = {} by Th56;
  reconsider XSeq = ASeq as SetSequence of Omega;
A3: (for k st k in dom FSeq holds XSeq.k = FSeq.k) &
  for k st not k in dom FSeq holds XSeq.k = {} by A1,A2;
  then P.(Union ASeq) = P.(Union FSeq) by Th55;
  then P.(Union FSeq) <= Sum(P * ASeq) by A1,A2,Th64;
  hence P.(Union FSeq) <= Sum(P * FSeq) by A1,A2,Th64;
  assume
A4: FSeq is disjoint_valued;
A5: FSeq is disjoint_valued implies ASeq is disjoint_valued
  proof
    assume
A6: FSeq is disjoint_valued;
    for m,n being Nat st m <> n holds ASeq.m misses ASeq.n
    proof
      let m,n be Nat such that
A7:   m <> n;
      per cases;
      suppose
A8:     m in dom FSeq & n in dom FSeq;
        FSeq.m misses FSeq.n by A6,A7,PROB_2:def 2;
        then ASeq.m misses FSeq.n by A1,A8;
        hence thesis by A1,A8;
      end;
      suppose
        not (m in dom FSeq & n in dom FSeq);
        then ASeq.m = {} or ASeq.n = {} by A2;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  thus P.(Union FSeq) = P.(Union ASeq) by Th55,A3
    .= Sum(P * ASeq) by A5,A4,Th46
    .= Sum(P * FSeq) by A1,A2,Th64;
end;
