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 Th8:
  (for A being set st A in rng ASeq holds P.A = 0) iff P.(union rng ASeq) = 0
proof
  hereby
    assume
A1: for A being set st A in rng ASeq holds P.A = 0;
    for n holds P.(ASeq.n) = 0
    by SETLIM_1:4,A1;
    then P.(Union ASeq) = 0 by Th7;
    hence P.(union rng ASeq) = 0 by CARD_3:def 4;
  end;
  assume P.(union rng ASeq) = 0;
  then
A2: P.(Union ASeq) = 0 by CARD_3:def 4;
  hereby
    let A be set;
    assume A in rng ASeq;
    then consider n1 being Nat such that
A3:  A = ASeq.n1 by SETLIM_1:4;
     n1 in NAT by ORDINAL1:def 12;
    hence P.A = 0 by A2,Th7,A3;
  end;
end;
