reserve
  A,B,X for set,
  S for SigmaField of X;
reserve C for C_Measure of X;

theorem Th12:
  sigma_Field(C) is SigmaField of X
proof
A1: C is nonnegative by Def1;
A2: for M being N_Sub_set_fam of X holds M c= sigma_Field(C) implies union M
  in sigma_Field(C)
  proof
    let M be N_Sub_set_fam of X;
    assume
A3: M c= sigma_Field(C);
    for W,Z being Subset of X holds (W c= union M & Z c= X \ union M
    implies C.W + C.Z <= C.(W \/ Z))
    proof
      reconsider S = bool X as SigmaField of X by Th3;
      let W,Z be Subset of X;
      assume that
A4:   W c= union M and
A5:   Z c= X \ union M;
      consider F being sequence of bool X such that
A6:   rng F = M by SUPINF_2:def 8;
      consider G being sequence of S such that
A7:   G.0 = F.0 and
A8:   for n being Nat holds G.(n+1) = F.(n+1) \/ G.n by MEASURE2:4;
      consider B being sequence of S such that
A9:   B.0 = F.0 and
A10:  for n being Nat holds B.(n+1) = F.(n+1) \ G.n by MEASURE2:8;
A11:  union rng F = union rng B by A7,A8,A9,A10,Th2;
      defpred P[Nat] means G.$1 in sigma_Field(C);
A12:  for n being Element of NAT holds F.n in sigma_Field(C)
      proof
        let n be Element of NAT;
        F.n in M by A6,FUNCT_2:4;
        hence thesis by A3;
      end;
A13:  for k being Nat st P[k] holds P[k+1]
      proof
        let k be Nat;
        assume
A14:    G.k in sigma_Field(C);
A15:    F.(k+1) in sigma_Field(C) by A12;
        G.(k+1) = F.(k+1) \/ G.k by A8;
        hence thesis by A14,A15,Th8;
      end;
A16:  P[0] by A12,A7;
A17:  for n being Nat holds P[n] from NAT_1:sch 2(A16,A13);
      consider Q being sequence of S such that
A18:  for n being Element of NAT holds Q.n = W /\ B.n by Th11;
A19:  union rng Q = W /\ union rng B by A18,Th1;
      consider QQ being sequence of S such that
A20:  QQ.0 = Q.0 and
A21:  for n being Nat holds QQ.(n+1) = Q.(n+1) \/ QQ.n by MEASURE2:4;
      reconsider Q,QQ,F,G as sequence of bool X;
A22:  F.0 in sigma_Field(C) by A12;
      defpred P[Nat] means C.(Z \/ QQ.$1) = C.Z + Ser(C*Q).$1;
A23:  C*Q is nonnegative by A1,MEASURE1:25;
A24:  for k being Nat st P[k] holds P[k+1]
      proof
        defpred P[Nat] means QQ.$1 c= G.$1;
        let k be Nat;
A25:    F.(k+1) \ G.k c= F.(k+1) by XBOOLE_1:36;
A26:    QQ.(k+1) = QQ.k \/ Q.(k+1) by A21;
A27:    G.k in sigma_Field(C) by A17;
        F.(k+1) in sigma_Field(C) by A12;
        then
A28:    F.(k+1) \ G.k in sigma_Field(C) by A27,Th10;
A29:    0. <= (C*Q).(k+1) by A23,SUPINF_2:39;
A30:    0. <= Ser(C*Q).k by A23,SUPINF_2:40;
        QQ.0 = W /\ F.0 by A9,A18,A20;
        then
A31:    P[0] by A7,XBOOLE_1:17;
        for n being Nat holds QQ.n misses (F.(n+1) \ G.n)
        proof
          let n be Nat;
A32:      for n being Nat st P[n] holds P[n+1]
          proof
            let n be Nat;
            assume
A33:        QQ.n c= G.n;
A34:        W /\ F.(n+1) c= F.(n+1) by XBOOLE_1:17;
            W /\ (F.(n+1) \ G.n) c= W /\ F.(n+1) by XBOOLE_1:26,36;
            then
A35:        W /\ (F.(n+1) \ G.n) c= F.(n+1) by A34;
            QQ.(n+1) = Q.(n+1) \/ QQ.n by A21
              .= (W /\ B.(n+1)) \/ QQ.n by A18
              .= (W /\ (F.(n+1) \ G.n)) \/ QQ.n by A10;
            then QQ.(n+1) c= F.(n+1) \/ G.n by A33,A35,XBOOLE_1:13;
            hence thesis by A8;
          end;
A36:      for n being Nat holds P[n] from NAT_1:sch 2(A31,A32);
          G.n misses (F.(n+1) \ G.n) by XBOOLE_1:79;
          hence thesis by A36,XBOOLE_1:63;
        end;
        then QQ.k misses (F.(k+1) \ G.k);
        then
A37:    QQ.k /\ (F.(k+1) \ G.k) = {};
A38:    QQ.k c= X \ (F.(k+1) \ G.k)
        proof
          let z be object;
          assume
A39:      z in QQ.k;
          then not z in F.(k+1) \ G.k by A37,XBOOLE_0:def 4;
          hence thesis by A39,XBOOLE_0:def 5;
        end;
        Q.(k+1) = W /\ B.(k+1) by A18
          .= W /\ (F.(k+1) \ G.k) by A10;
        then
A40:    Q.(k+1) c= F.(k+1) \ G.k by XBOOLE_1:17;
        F.(k+1) c= union rng F by FUNCT_2:4,ZFMISC_1:74;
        then F.(k+1) \ G.k c= union rng F by A25;
        then X \ union rng F c= X \ (F.(k+1) \ G.k) by XBOOLE_1:34;
        then Z c= X \ (F.(k+1) \ G.k) by A5,A6;
        then Z \/ QQ.k c= X \ (F.(k+1) \ G.k) by A38,XBOOLE_1:8;
        then
A41:    C.(Q.(k+1)) + C.(Z \/ QQ.k) = C.((Z \/ QQ.k) \/ Q.(k+1)) by A40,A28,Th5
          .= C.(Z \/ QQ.(k+1)) by A26,XBOOLE_1:4;
A42:    0. <= C.Z by A1,SUPINF_2:39;
        assume C.(Z \/ QQ.k) = C.Z + Ser(C*Q).k;
        then C.(Z \/ QQ.(k+1)) = (C.Z + Ser(C*Q).k) + (C*Q).(k+1) by A41,
FUNCT_2:15
          .= C.Z + (Ser(C*Q).k + (C*Q).(k+1)) by A42,A30,A29,XXREAL_3:44
          .= C.Z + Ser(C*Q).(k+1) by SUPINF_2:def 11;
        hence thesis;
      end;
      QQ.0 = W /\ F.0 by A9,A18,A20;
      then
A43:  QQ.0 c= F.0 by XBOOLE_1:17;
A44:  Ser(C*Q).0 = (C*Q).0 by SUPINF_2:def 11
        .= C.(QQ.0) by A20,FUNCT_2:15;
      F.0 in rng F by FUNCT_2:4;
      then X \ union rng F c= X \ F.0 by XBOOLE_1:34,ZFMISC_1:74;
      then Z c= X \ F.0 by A5,A6;
      then
A45:  P[0] by A22,A43,A44,Th5;
A46:  for n being Nat holds P[n] from NAT_1:sch 2(A45,A24);
      defpred Q[Nat] means QQ.$1 c= W;
A47:  for n being Nat st Q[n] holds Q[n+1]
      proof
        let n be Nat;
        assume
A48:    QQ.n c= W;
A49:    W /\ B.(n+1) c= W by XBOOLE_1:17;
        QQ.(n+1) = Q.(n+1) \/ QQ.n by A21
          .= (W /\ B.(n+1)) \/ QQ.n by A18;
        then QQ.(n+1) c= W \/ W by A48,A49,XBOOLE_1:13;
        hence thesis;
      end;
      QQ.0 = W /\ B.0 by A18,A20;
      then
A50:  Q[0] by XBOOLE_1:17;
A51:  for n being Nat holds Q[n] from NAT_1:sch 2(A50,A47);
A52:  union rng Q = W by A4,A6,A11,A19,XBOOLE_1:28;
A53:  C.Z is real implies C.W + C.Z <= C.(W \/ Z )
      proof
         defpred P[object,object] means
($1 = 0 implies $2 = C.(Z \/ W) - C.Z ) & ($1
        <> 0 implies $2 = 0.);
A54:    for x being object st x in NAT
          ex y being object st y in ExtREAL & P[x, y]
        proof
          let x be object;
          assume x in NAT;
          then reconsider x as Element of NAT;
          x <> 0 implies ex y being set st y in ExtREAL & P[x,y];
          hence thesis;
        end;
        consider R being sequence of ExtREAL such that
A55:    for x being object st x in NAT holds P[x,R.x] from FUNCT_2:sch 1
        (A54);
        assume
A56:    C.Z is real;
        for n being Element of NAT holds 0. <= R.n
        proof
          let n be Element of NAT;
            C.Z in REAL or C.Z in {-infty,+infty} by XBOOLE_0:def 3
,XXREAL_0:def 4;
            then consider y being Element of REAL such that
A57:        y = C.Z by A56,TARSKI:def 2;
            Z c= Z \/ W by XBOOLE_1:7;
            then C.Z <= C.(Z \/ W) by Def1;
            then
A58:        C.Z - C.Z <= C.(Z \/ W) - C.Z by XXREAL_3:37;
            C.Z - C.Z = y - y by A57,SUPINF_2:3;
            hence thesis by A55,A58;
        end;
        then
A59:    R is nonnegative by SUPINF_2:39;
A60:    for n being Element of NAT holds Ser(C*Q).n <= C.(Z \/ W) - C.Z
        proof
          let n be Element of NAT;
A61:      Z \/ QQ.n c= Z \/ W by A51,XBOOLE_1:9;
          Ser(C*Q).n + C.Z = C.(Z \/ QQ.n) by A46;
          then Ser(C*Q).n + C.Z <= C.(Z \/ W) by A61,Def1;
          hence thesis by A56,XXREAL_3:45;
        end;
A62:    for n being Element of NAT holds Ser(C*Q).n <= Ser(R).n
        proof
          let n be Element of NAT;
          defpred P[Nat] means Ser(R).$1 = C.(Z \/ W) - C.Z;
A63:      for k being Nat st P[k] holds P[k+1]
          proof
            let k be Nat;
            assume
A64:        Ser(R).k = C.(Z \/ W) - C.Z;
            set y = Ser(R).k;
            thus Ser(R).(k+1) = y + R.(k+1) by SUPINF_2:def 11
              .= y + 0. by A55
              .= C.(Z \/ W) - C.Z by A64,XXREAL_3:4;
          end;
          Ser(R).0 = R.0 by SUPINF_2:def 11;
          then
A65:      P[0] by A55;
          for k being Nat holds P[k] from NAT_1:sch 2(A65,A63);
          then Ser(R).n = C.(Z \/ W) - C.Z;
          hence thesis by A60;
        end;
        set y = Ser(R).0;
        y = R.0 by SUPINF_2:def 11;
        then
A66:    y = C.(Z \/ W) - C.Z by A55;
A67:    C.W <= SUM(C*Q) by A52,Def1;
        for k being Element of NAT st 1 <= k holds R.k = 0. by A55;
        then
A68:    SUM(R) = Ser(R).1 by A59,SUPINF_2:48;
        Ser(R).1 = y + R.(0+1) by SUPINF_2:def 11
          .= y + 0. by A55
          .= C.(Z \/ W) - C.Z by A66,XXREAL_3:4;
        then SUM(C*Q) <= C.(Z \/ W) - C.Z by A62,A68,MEASURE3:1;
        then C.W <= C.(Z \/ W) - C.Z by A67,XXREAL_0:2;
        hence thesis by A56,XXREAL_3:45;
      end;
A69:  C.Z = +infty implies C.W + C.Z <= C.(W \/ Z)
      proof
A70:    Z c= W \/ Z by XBOOLE_1:7;
        assume
A71:    C.Z = +infty;
        0. <= C.W by A1,MEASURE1:def 2;
        then C.W + C.Z = +infty by A71,XXREAL_3:def 2;
        hence thesis by A71,A70,Def1;
      end;
      0 <= C.Z by A1,MEASURE1:def 2;
      then C.Z is Element of REAL or C.Z = +infty by XXREAL_0:14;
      hence thesis by A69,A53;
    end;
    hence thesis by Def2;
  end;
  for A being set holds A in sigma_Field(C) implies X\A in sigma_Field(C)
  by Th7;
  then reconsider Y = sigma_Field C as non empty compl-closed sigma-additive
  Subset-Family of X by A2,MEASURE1:def 1,def 5;
  Y is SigmaField of X;
  hence thesis;
end;
