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 Th14:
  for X being non empty set, S being SigmaField of X, M being
  sigma_Measure of S st M.X = 1 holds M is Probability of S
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S such
  that
A1: M.X = 1;
A2: for A being Element of S holds M.A <= 1
  proof
    reconsider X as Element of S by PROB_1:5;
    let A be Element of S;
    M.A <= M.X by MEASURE1:31;
    hence thesis by A1;
  end;
A3: for x being object st x in S holds M.x in REAL
  proof
    let x be object;
    assume x in S;
    then reconsider x as Element of S;
A4: 1 in REAL & 0 in REAL by XREAL_0:def 1;
    0. <= M.x & M.x <= 1 by A2,MEASURE1:def 2;
    hence thesis by A4,XXREAL_0:45;
  end;
  dom M = S by FUNCT_2:def 1;
  then reconsider P1 = M as Function of S,REAL by A3,FUNCT_2:3;
  reconsider P1 as Function of S,REAL;
A5: for ASeq being SetSequence of S st ASeq is non-ascending holds P1 *
  ASeq is convergent & lim (P1 * ASeq) = P1.Intersection ASeq
  proof
    let ASeq be SetSequence of S such that
A6: ASeq is non-ascending;
    for n being Nat holds 0 <= (P1 * ASeq).n
    proof
      let n be Nat;
A7:    n in NAT by ORDINAL1:def 12;
      reconsider A = ASeq.n as Event of S;
      0 <= P1.A & dom (P1 * ASeq) = NAT by MEASURE1:def 2,SEQ_1:1;
      hence thesis by FUNCT_1:12,A7;
    end;
    then
A8: P1*ASeq is bounded_below by RINFSUP1:10;
    reconsider F = ASeq as sequence of S by Th2;
A9: for n being Nat holds F.(n+1) c= F.n by A6,PROB_2:6;
A10: M.(F.0) <+infty by A3,XXREAL_0:9;
    now
      let n be Nat;
       reconsider nn=n as Element of NAT by ORDINAL1:def 12;
      dom (M*F) = NAT by FUNCT_2:def 1;
      then
A11:  (M*F).nn = M.(F.nn) & (M*F).(nn+1) = M.(F.(nn+1)) by FUNCT_1:12;
      F.(n+1) c= F.n by A6,PROB_2:6;
      hence (P1*ASeq).(n+1) <= (P1*ASeq).n by A11,MEASURE1:31;
    end;
    then
A12: P1*ASeq is non-increasing by SEQM_3:def 9;
    then lim (P1*ASeq) = lower_bound (P1*ASeq) by A8,RINFSUP1:25
      .= inf(rng (M*F)) by A8,Th11;
    then lim (P1 * ASeq) = M.(meet rng F) by A9,A10,MEASURE3:12
      .= P1.Intersection ASeq by SETLIM_1:8;
    hence thesis by A8,A12;
  end;
A13: for A,B being Event of S st A misses B holds P1.(A \/ B) = P1.A + P1.B
  proof
    let A,B be Event of S such that
A14: A misses B;
    reconsider A, B as Element of S;
    reconsider A2 = A, B2 = B as Element of S;
    P1.(A \/ B) = M.A2 + M.B2 by A14,MEASURE1:30
      .= P1.A + P1.B by SUPINF_2:1;
    hence thesis;
  end;
  ( for A being Event of S holds 0 <= P1.A)& P1.X = 1 by A1,MEASURE1:def 2;
  hence thesis by A13,A5,PROB_1:def 8;
end;
