reserve X for set;

theorem Th13:
  for S being SigmaField of X, M being Measure of S, F being
  Sep_Sequence of S holds SUM(M*F) <= M.(union rng F)
proof
  let S be SigmaField of X, M be Measure of S, F be Sep_Sequence of S;
  set T = rng F;
  consider G being sequence of S such that
A1: G.0 = F.0 and
A2: for n being Nat holds G.(n+1) = F.(n+1) \/ G.n by MEASURE2:4;
  {} is Subset of X by XBOOLE_1:2;
  then consider H being sequence of bool X such that
A3: rng H = {union T,{}} and
A4: H.0 = union T and
A5: for n being Nat st 0 < n holds H.n = {} by MEASURE1:19;
  rng H c= S
  proof
    let a be object;
    assume a in rng H;
    then a = union T or a = {} by A3,TARSKI:def 2;
    hence thesis by PROB_1:4;
  end;
  then reconsider H as sequence of S by FUNCT_2:6;
  defpred P[Nat] means Ser(M*F).$1 = M.(G.$1);
A6: dom (M*F) = NAT by FUNCT_2:def 1;
A7: for n being Nat holds G.n /\ F.(n+1) = {}
  proof
    let n be Nat;
A8: for n being Nat holds for k being Element of NAT st n < k
    holds G.n /\ F.k = {}
    proof
      defpred P[Nat] means
   for k being Element of NAT st $1 < k
      holds G.$1 /\ F.k = {};
A9:   for n being Nat st P[n] holds P[n+1]
      proof
        let n be Nat;
        assume
A10:    for k being Element of NAT st n < k holds G.n /\ F.k = {};
        let k be Element of NAT;
        assume
A11:    n+1 < k;
        then
A12:    n < k by NAT_1:13;
        F.(n+1) misses F.k by A11,PROB_2:def 2;
        then
A13:    F.(n+1) /\ F.k = {};
        G.(n+1) /\ F.k = (F.(n+1) \/ G.n) /\ F.k by A2
          .= (F.(n+1) /\ F.k) \/ (G.n /\ F.k) by XBOOLE_1:23;
        hence thesis by A10,A12,A13;
      end;
A14:  P[0]
      by PROB_2:def 2,A1,XBOOLE_0:def 7;
      thus for n being Nat holds P[n] from NAT_1:sch 2(A14, A9);
    end;
    n < n + 1 by NAT_1:13;
    hence thesis by A8;
  end;
A15: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    G.k /\ F.(k+1) = {} by A7;
    then
A16: G.k misses F.(k+1);
    assume Ser(M*F).k = M.(G.k);
    then Ser(M*F).(k+1) = M.(G.k) + (M*F).(k+1) by SUPINF_2:def 11;
    then Ser(M*F).(k+1) = M.(G.k) + M.(F.(k+1)) by A6,FUNCT_1:12
      .= M.(F.(k+1) \/ G.k) by A16,MEASURE1:def 3
      .= M.(G.(k+1)) by A2;
    hence thesis;
  end;
  Ser(M*F).0 = (M*F).0 by SUPINF_2:def 11;
  then
A17: P[0] by A1,A6,FUNCT_1:12;
A18: for n being Nat holds P[n] from NAT_1:sch 2(A17,A15 );
  defpred P[Nat] means Ser(M*H).$1 = M.(union T);
A19: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    0 <= n by NAT_1:2;
    then 0 < n + 1 by NAT_1:13;
    then
A20: H.(n+1) = {} by A5;
    dom (M*H) = NAT by FUNCT_2:def 1;
    then (M*H).(n+1) = M.({}) by A20,FUNCT_1:12;
    then
A21: (M*H).(n+1) = 0. by VALUED_0:def 19;
    assume Ser(M*H).n = M.(union T);
    then Ser(M*H).(n+1) = M.(union T) + (M*H).(n+1) by SUPINF_2:def 11;
    hence thesis by A21,XXREAL_3:4;
  end;
  Ser(M*H).0 = (M*H).0 & dom (M*H) = NAT by FUNCT_2:def 1,SUPINF_2:def 11;
  then
A22: P[0] by A4,FUNCT_1:12;
A23: for n being Nat holds P[n] from NAT_1:sch 2(A22,A19 );
A24: for r being Element of NAT st 1 <= r holds (M*H).r = 0.
  proof
    let r be Element of NAT;
    assume 1 <= r;
    then 0 + 1 <= r;
    then 0 < r by NAT_1:13;
    then
A25: H.r = {} by A5;
    dom (M*H) = NAT by FUNCT_2:def 1;
    then (M*H).r = M.({}) by A25,FUNCT_1:12;
    hence thesis by VALUED_0:def 19;
  end;
A26: for n being Nat holds G.n c= union T
  proof
    defpred P[Nat] means G.$1 c= union T;
A27: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume
A28:  G.n c= union T;
      G.(n+1) = F.(n+1) \/ G.n & F.(n+1) c= union T by A2,FUNCT_2:4,ZFMISC_1:74
;
      hence thesis by A28,XBOOLE_1:8;
    end;
A29: P[0] by A1,FUNCT_2:4,ZFMISC_1:74;
    thus for n being Nat holds P[n] from NAT_1:sch 2(A29,A27 );
  end;
A30: for n being Element of NAT holds Ser(M*F).n <= Ser(M*H).n
  proof
    let n be Element of NAT;
    Ser(M*F).n = M.(G.n) by A18;
    then Ser(M*F).n <= M.(union T) by A26,MEASURE1:8;
    hence thesis by A23;
  end;
  M*H is nonnegative by MEASURE1:25;
  then SUM(M*H) = Ser(M*H).1 by A24,SUPINF_2:48;
  then SUM(M*H) = M.(union T) by A23;
  hence thesis by A30,Th1;
end;
