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 Th43:
  ASeq is disjoint_valued implies (P * Partial_Union ASeq).n =
  Partial_Sums(P * ASeq).n
proof
A1: dom (P * ASeq) = NAT by SEQ_1:1;
  defpred P[Nat] means (P * Partial_Union ASeq).$1 = Partial_Sums(P * ASeq).
  $1;
A2: dom (P * Partial_Union ASeq) = NAT by SEQ_1:1;
  assume
A3: ASeq is disjoint_valued;
A4: for k st P[k] holds P[k+1]
  proof
    let k such that
A5: (P * Partial_Union ASeq).k = Partial_Sums(P * ASeq).k;
    k < k+1 by NAT_1:13;
    then
A6: (Partial_Union ASeq).k misses ASeq.(k+1) by A3,Th42;
    reconsider k as Element of NAT by ORDINAL1:def 12;
A7: Partial_Sums(P * ASeq).(k+1) =Partial_Sums(P * ASeq).k + (P * ASeq).(
    k+1) by SERIES_1:def 1
      .=Partial_Sums(P * ASeq).k + P.(ASeq.(k+1)) by A1,FUNCT_1:12;
    (P * Partial_Union ASeq).(k+1) = P.((Partial_Union ASeq).(k+1)) by A2,
FUNCT_1:12
      .= P.((Partial_Union ASeq).k \/ ASeq.(k+1)) by Def2
      .= P.((Partial_Union ASeq).k) + P.(ASeq.(k+1)) by A6,PROB_1:def 8
      .= (P * Partial_Union ASeq).k + P.(ASeq.(k+1)) by A2,FUNCT_1:12;
    hence thesis by A5,A7;
  end;
A8: P[0] by Th40;
  for k holds P[k] from NAT_1:sch 2(A8,A4);
  hence thesis;
end;
