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 Th40:
  (P * Partial_Union ASeq).0 = Partial_Sums(P * ASeq).0
proof
A1: dom (P * ASeq) = NAT by SEQ_1:1;
  dom (P * Partial_Union ASeq) = NAT by SEQ_1:1;
  then
A2: (P * Partial_Union ASeq).0 = P.((Partial_Union ASeq).0) by FUNCT_1:12
    .= P.(ASeq.0) by Def2;
  Partial_Sums(P * ASeq).0 = (P * ASeq).0 by SERIES_1:def 1
    .= P.(ASeq.0) by A1,FUNCT_1:12;
  hence thesis by A2;
end;
