reserve Omega for set;
reserve m,n,k for Nat;
reserve x,y for object;
reserve r,r1,r2,r3 for Real;
reserve seq,seq1 for Real_Sequence;
reserve Sigma for SigmaField of Omega;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve P,P1,P2 for Probability of Sigma;

theorem
 :: Equivalent Definition of Probability
  for P being Function of Sigma,REAL holds P is Probability of Sigma iff
(for A holds 0 <= P.A) & P.Omega = 1 & (for A,B st A misses B holds P.(A \/ B)
= P.A + P.B) & for ASeq st ASeq is non-descending holds P * ASeq is convergent
  & lim (P * ASeq) = P.Union ASeq
proof
  let P be Function of Sigma,REAL;
  thus P is Probability of Sigma implies (for A holds 0 <= P.A) & P.Omega = 1
  & (for A,B st A misses B holds P.(A \/ B) = P.A + P.B) & for ASeq st ASeq is
non-descending holds P * ASeq is convergent & lim (P * ASeq) = P.Union ASeq by
Lm2,PROB_1:def 8;
  assume that
A1: for A holds 0 <= P.A and
A2: P.Omega = 1 and
A3: for A,B st A misses B holds P.(A \/ B) = P.A + P.B and
A4: for ASeq st ASeq is non-descending holds P * ASeq is convergent &
  lim (P * ASeq) = P.Union ASeq;
  for ASeq st ASeq is non-ascending holds P * ASeq is convergent & lim (P
  * ASeq) = P.Intersection ASeq
  proof
    let ASeq such that
A5: ASeq is non-ascending;
    Intersection ASeq = @Intersection ASeq;
    then reconsider V = Intersection ASeq as Event of Sigma;
    set BSeq = Complement ASeq;
A6: for A holds P.(A`) = 1 - P.A
    proof
      let A;
      reconsider B = A` as Event of Sigma by PROB_1:20;
A7:   A misses B by SUBSET_1:24;
      1 = P.[#]Omega by A2
        .= P.(A \/ B) by SUBSET_1:10
        .= P.A + P.B by A3,A7;
      hence thesis;
    end;
A8: now
      let n;
       reconsider nn=n as Element of NAT by ORDINAL1:def 12;
      (P * BSeq).n = P.(BSeq.nn) by FUNCT_2:15
        .= P.((ASeq.n)`) by PROB_1:def 2
        .= 1 - P.(ASeq.n) by A6
        .= 1 - (P * ASeq).nn by FUNCT_2:15
        .= 1 + - (P * ASeq).n;
      hence (P * ASeq).n = 1 - (P * BSeq).n;
    end;
    Union BSeq = (Intersection ASeq)`;
    then
A9: P.Union BSeq = 1 - P.V by A6;
A10: BSeq is non-descending by A5,Th8;
    then
A11: P * BSeq is convergent by A4;
    hence P * ASeq is convergent by A8,Th2;
    thus lim (P * ASeq) = 1 - lim (P * BSeq) by A11,A8,Th2
      .= 1 - (1 - P.V) by A4,A10,A9
      .= P.(Intersection ASeq);
  end;
  hence thesis by A1,A2,A3,PROB_1:def 8;
end;
