reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;
reserve FSets for Set_Sequence of F,
  CA for Covering of A,F;
reserve Cvr for Covering of Sets,F;

theorem Th13:
  (C_Meas M).(union rng Sets) <= SUM((C_Meas M)*Sets)
proof
A1: now
    assume
A2: for n being Element of NAT holds Svc(M,Sets.n) <> {+infty};
    inf Svc(M,union rng Sets) <= sup rng Ser((C_Meas M)*Sets)
    proof
      set y = inf Svc(M,union rng Sets), x = sup rng Ser((C_Meas M)*Sets);
A3:   Ser((C_Meas M)*Sets).0 <= x by FUNCT_2:4,XXREAL_2:4;
A4:   (C_Meas M)*Sets is nonnegative by Th10,MEASURE1:25;
      then 0 <= ((C_Meas M)*Sets).0 by SUPINF_2:39;
      then
A5:   0 <= Ser((C_Meas M)*Sets).0 by SUPINF_2:def 11;
      assume not inf Svc(M,union rng Sets) <= sup rng Ser((C_Meas M)*Sets);
      then consider eps being Real such that
A6:   0 < eps and
A7:   x + eps < y by A5,A3,XXREAL_3:49;
      consider E being sequence of ExtREAL such that
A8:   for n being Nat holds 0 < E.n and
A9:   SUM E < eps by A6,MEASURE6:4;
      for n being Element of NAT holds 0 <= E.n by A8;
      then
A10:  E is nonnegative by SUPINF_2:39;
      defpred P[Element of NAT,set] means ex F0 being Covering of Sets.$1,F st
      $2 = F0 & SUM vol(M,F0) < inf Svc(M,Sets.$1) + E.$1;
A11:  for n being Element of NAT holds ex F0 being Element of Funcs(NAT,
      bool X) st P[n,F0]
      proof
        let n be Element of NAT;
        C_Meas M is nonnegative & (C_Meas M).(Sets.n) = inf Svc(M,Sets.n)
        by Def8,Th10;
        then
A12:    0 in REAL & 0. <= inf Svc(M,Sets.n) by SUPINF_2:51,XREAL_0:def 1;
        Svc(M,Sets.n) <> {+infty} by A2;
        then not Svc(M,Sets.n) c= {+infty} by ZFMISC_1:33;
        then Svc(M,Sets.n) \ {+infty} <> {} by XBOOLE_1:37;
        then consider x being object such that
A13:    x in Svc(M,Sets.n) \ {+infty} by XBOOLE_0:def 1;
        reconsider x as R_eal by A13;
        not x in {+infty} by A13,XBOOLE_0:def 5;
        then
A14:    x <> +infty by TARSKI:def 1;
        x in Svc(M,Sets.n) by A13,XBOOLE_0:def 5;
        then inf Svc(M,Sets.n) <= x by XXREAL_2:3;
        then inf Svc(M,Sets.n) < +infty by A14,XXREAL_0:2,4;
        then inf Svc(M,Sets.n) is Element of REAL by A12,XXREAL_0:46;
        then consider S1 being ExtReal such that
A15:    S1 in Svc(M,Sets.n) and
A16:    S1 < inf Svc(M,Sets.n) + E.n by A8,MEASURE6:5;
        consider FS being Covering of Sets.n,F such that
A17:    S1 = SUM vol(M,FS) by A15,Def7;
        reconsider FS as Element of Funcs(NAT,bool X) by FUNCT_2:8;
        take FS;
        thus thesis by A16,A17;
      end;
      consider FF being sequence of Funcs(NAT,bool X) such that
A18:  for n being Element of NAT holds P[n,FF.n] from FUNCT_2:sch 3(
      A11);
A19:  for n being Nat holds FF.n is Covering of Sets.n,F
      proof
        let n be Nat;
        n in NAT by ORDINAL1:def 12;
        then ex F0 being Covering of Sets.n,F st F0 = FF.n & SUM vol(M,F0) <
        inf Svc(M,Sets.n) + E.n by A18;
        hence thesis;
      end;
      deffunc F(Element of NAT) = ((C_Meas M)*Sets).$1 + E.$1;
A20:  for x being Element of NAT holds F(x) is Element of ExtREAL;
      consider G0 being sequence of ExtREAL such that
A21:  for n being Element of NAT holds G0.n = F(n) from FUNCT_2:sch 9
      (A20);
      reconsider FF as Covering of Sets,F by A19,Def4;
      for n being Element of NAT holds (Volume(M,FF)).n <= G0.n
      proof
        let n be Element of NAT;
        (ex F0 being Covering of Sets.n,F st F0 = FF.n & SUM vol( M,F0) <
inf Svc(M, Sets.n) + E.n )& ((C_Meas M)*Sets).n = (C_Meas M).(Sets.n) by A18,
FUNCT_2:15;
        then SUM vol(M,FF.n) < ((C_Meas M)*Sets).n + E.n by Def8;
        then (Volume(M,FF)).n < ((C_Meas M)*Sets).n + E.n by Def6;
        hence thesis by A21;
      end;
      then
A22:  SUM Volume(M,FF) <= SUM G0 by SUPINF_2:43;
A23:  now
        let n be Nat;
        n is Element of NAT by ORDINAL1:def 12;
        hence G0.n = ((C_Meas M)*Sets).n + E.n by A21;
      end;
      SUM((C_Meas M)*Sets) + SUM(E) <= SUM((C_Meas M)*Sets) + eps by A9,
XXREAL_3:36;
      then SUM G0 <= SUM((C_Meas M)*Sets) + eps by A4,A10,A23,Th3;
      then
A24:  SUM Volume(M,FF) <= SUM((C_Meas M)*Sets) + eps by A22,XXREAL_0:2;
      y <= SUM Volume(M,FF) by Th7;
      hence thesis by A7,A24,XXREAL_0:2;
    end;
    hence thesis by Def8;
  end;
  now
    assume ex n being Element of NAT st Svc(M,Sets.n) = {+infty};
    then consider n being Element of NAT such that
A25: Svc(M,Sets.n) = {+infty};
    inf {+infty} = +infty by XXREAL_2:13;
    then (C_Meas M).(Sets.n) = +infty by A25,Def8;
    hence thesis by Lm2;
  end;
  hence thesis by A1;
end;
