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 Th41:
  P * Partial_Union ASeq is convergent & lim (P * Partial_Union
ASeq) = upper_bound (P * Partial_Union ASeq) & lim (P * Partial_Union ASeq)
 = P.Union
  ASeq
proof
A1: P * Partial_Union ASeq is non-decreasing by Th37;
A2: Partial_Union ASeq is non-descending by Th11;
  then P * Partial_Union ASeq is convergent by PROB_2:10;
  then
A3: P * Partial_Union ASeq is bounded;
  lim (P * Partial_Union ASeq) = P.Union (Partial_Union ASeq) by A2,PROB_2:10
    .= P.Union ASeq by Th15;
  hence thesis by A2,A3,A1,PROB_2:10,RINFSUP1:24;
end;
