reserve n,m,k for Element of NAT,
  x,X for set,
  A1 for SetSequence of X,
  Si for SigmaField of X,
  XSeq for SetSequence of Si;
reserve Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th7:
  (for n holds P.(ASeq.n) = 0) iff P.(Union ASeq) = 0
proof
  hereby
    assume
A1: for n holds P.(ASeq.n) = 0;
    for n being Nat holds Partial_Sums(P * ASeq).n = 0
    proof
      defpred P[Nat] means Partial_Sums(P * ASeq).$1 = 0;
A2:   for k being Nat st P[k] holds P[k+1]
      proof
        let k be Nat such that
A3:     P[k];
        thus Partial_Sums(P * ASeq).(k+1) = Partial_Sums(P * ASeq).k + (P *
        ASeq).(k+1) by SERIES_1:def 1
          .= 0 + P.(ASeq.(k+1)) by A3,FUNCT_2:15
          .= 0 by A1;
      end;
      Partial_Sums(P * ASeq).0 = (P * ASeq).0 by SERIES_1:def 1
        .= P.(ASeq.0) by FUNCT_2:15
        .= 0 by A1;
      then
A4:   P[0];
      thus for k being Nat holds P[k] from NAT_1:sch 2(A4,A2);
    end;
    then for n being Nat st 0 <= n holds Partial_Sums(P * ASeq).n = 0;
    then
A5: Partial_Sums(P * ASeq) is convergent & lim Partial_Sums(P * ASeq) = 0
    by PROB_1:1;
    now
      let n be Nat;
      (Partial_Diff_Union ASeq).n c= ASeq.n by PROB_3:33;
      hence (P * Partial_Diff_Union ASeq).n <= (P * ASeq).n by PROB_3:5;
    end;
    then
A6: for n being Nat
     holds (Partial_Sums(P * Partial_Diff_Union ASeq)).n <= (
    Partial_Sums(P * ASeq)).n by SERIES_1:14;
    Partial_Sums(P * Partial_Diff_Union ASeq) is convergent by PROB_3:45;
    then Sum (P * Partial_Diff_Union ASeq) <= 0 by A5,A6,SEQ_2:18;
    then P.(Union Partial_Diff_Union ASeq) <= 0 by PROB_3:46;
    then
A7: P.(Union ASeq) <= 0 by PROB_3:36;
    Union ASeq is Event of Sigma by PROB_1:26;
    hence P.(Union ASeq) = 0 by A7,PROB_1:def 8;
  end;
  assume
A8: P.(Union ASeq) = 0;
  hereby
    reconsider Y2 = Union ASeq as Event of Sigma by PROB_1:26;
    let n;
    reconsider Y1 = ASeq.n as Event of Sigma;
    ASeq.n in rng ASeq by SETLIM_1:4;
    then ASeq.n c= union rng ASeq by ZFMISC_1:74;
    then Y1 c= Y2 by CARD_3:def 4;
    hence P.(ASeq.n) = 0 by A8,Th6;
  end;
end;
