 reserve Omega for non empty set;
 reserve r for Real;
 reserve Sigma for SigmaField of Omega;
 reserve P for Probability of Sigma;

theorem Th8:
  for Omega be non empty finite set, f be Function of Omega,REAL,
  P be Function of bool Omega, REAL st
  (for x be set st x c= Omega holds 0 <= P.x & P.x <= 1) &
  P.Omega = 1 & for z be finite Subset of Omega
  holds P.z = setopfunc(z,Omega,REAL,f,addreal)
  holds P is Probability of Trivial-SigmaField Omega
  proof
    let Omega be non empty finite set,
    f be Function of Omega,REAL,
    P be Function of bool Omega, REAL;
    assume that
    A1: for x be set st x c= Omega holds 0 <= P.x & P.x <= 1 and
    A2: P.Omega = 1 and
    A3: for z be finite Subset of Omega
    holds P.z = setopfunc(z,Omega,REAL,f,addreal);
    A4: for A,B being Event of Trivial-SigmaField (Omega)
    st A misses B holds P.(A \/ B) = P.A + (P.B qua Real)
    proof
      let A,B be Event of Trivial-SigmaField (Omega);
      assume
      A5: A misses B;
      reconsider A0 =A,B0=B as finite Subset of Omega;
      A6: Omega = dom f by FUNCT_2:def 1;
      thus P.(A \/ B) = setopfunc((A0 \/ B0),Omega,REAL,f,addreal) by A3
      .= (addreal).(setopfunc(A0,Omega,REAL,f,addreal),
      setopfunc(B0,Omega,REAL,f,addreal)) by A5,A6,BHSP_5:14
      .= (addreal).(setopfunc(A0,Omega,REAL,f,addreal),(P.B)) by A3
      .= (addreal).((P.A),(P.B)) by A3
      .= (P.A) + ((P.B) qua Real) by BINOP_2:def 9;
    end;
    A7: for A being Event of Trivial-SigmaField (Omega) holds 0 <= P.A by A1;
     for ASeq being SetSequence of Trivial-SigmaField (Omega) st ASeq is
    non-ascending holds P * ASeq is convergent & lim (P * ASeq) = P.(
    Intersection ASeq)
    proof
      let ASeq being SetSequence of Trivial-SigmaField (Omega);
      assume ASeq is non-ascending;
      then consider N be Nat such that
      A8: for m be Nat st N<=m holds Intersection ASeq = ASeq.m
      by RANDOM_1:15;
      now
        let m be Nat;
        assume
        A9: N <= m;
        m in NAT by ORDINAL1:def 12;
        then m in dom ASeq by FUNCT_2:def 1;
        hence (P * ASeq).m = P.(ASeq.m) by FUNCT_1:13
        .= P.(Intersection ASeq) by A8,A9;
      end;
      hence thesis by PROB_1:1;
    end;
    hence thesis by A7,A4,A2,PROB_1:def 8;
  end;
