reserve X for set;

theorem Th11:
  for S being SigmaField of X, M being sigma_Measure of S, T being
N_Measure_fam of S, F being sequence of S st T = rng F holds M.(union T) <=
  SUM(M*F)
proof
  let S be SigmaField of X, M be sigma_Measure of S, T be N_Measure_fam of S,
  F be sequence of S;
  consider G being sequence of S such that
A1: G.0 = F.0 & for n being Nat holds G.(n+1) = F.(n+1) \/ G.
  n by Th4;
  consider H being sequence of S such that
A2: H.0 = F.0 and
A3: for n being Nat holds H.(n+1) = F.(n+1) \ G.n by Th8;
  for n,m being object st n <> m holds H.n misses H.m
  proof
    let n,m be object;
    assume
A4: n <> m;
    per cases;
    suppose
      n in dom H & m in dom H;
      then reconsider n9=n,m9=m as Element of NAT;
A5:   m9 < n9 implies H.m misses H.n by A1,A2,A3,Th10;
      n9 < m9 implies H.n misses H.m by A1,A2,A3,Th10;
      hence thesis by A4,A5,XXREAL_0:1;
    end;
    suppose
      not (n in dom H & m in dom H);
      then H.n = {} or H.m = {} by FUNCT_1:def 2;
      hence thesis;
    end;
  end;
  then H is Sep_Sequence of S by PROB_2:def 2;
  then
A6: SUM(M*H) = M.(union rng H) by MEASURE1:def 6;
A7: for n being Element of NAT holds H.n c= F.n
  proof
    let n be Element of NAT;
A8: (ex k being Nat st n = k + 1) implies H.n c= F.n
    proof
      given k being Nat such that
A9:   n = k + 1;
      reconsider k as Element of NAT by ORDINAL1:def 12;
      H.n = F.n \ G.k by A3,A9;
      hence thesis by XBOOLE_1:36;
    end;
    n=0 or ex k being Nat st n = k + 1 by NAT_1:6;
    hence thesis by A2,A8;
  end;
A10: for n being Element of NAT holds (M*H).n <= (M*F).n
  proof
    let n be Element of NAT;
    NAT = dom(M*F) by FUNCT_2:def 1;
    then
A11: (M*F).n = M.(F.n) by FUNCT_1:12;
    NAT = dom(M*H) by FUNCT_2:def 1;
    then (M*H).n = M.(H.n) by FUNCT_1:12;
    hence thesis by A7,A11,MEASURE1:31;
  end;
A12: union rng H = union rng F
  proof
    thus union rng H c= union rng F
    proof
      let r be object;
      assume r in union rng H;
      then consider E being set such that
A13:  r in E and
A14:  E in rng H by TARSKI:def 4;
      consider s being object such that
A15:  s in dom H and
A16:  E = H.s by A14,FUNCT_1:def 3;
      reconsider s as Element of NAT by A15;
A17:  F.s in rng F by FUNCT_2:4;
      E c= F.s by A7,A16;
      hence thesis by A13,A17,TARSKI:def 4;
    end;
    let r be object;
    assume r in union rng F;
    then consider E being set such that
A18: r in E and
A19: E in rng F by TARSKI:def 4;
A20: ex s being object st s in dom F & E = F.s by A19,FUNCT_1:def 3;
    ex s1 being Element of NAT st r in H.s1
    proof
      defpred P[Nat] means r in F.$1;
A21:  ex k being Nat st P[k] by A18,A20;
      ex k being Nat st P[k] & for n being Nat st P[n] holds k <= n from
      NAT_1:sch 5(A21);
      then consider k being Nat such that
A22:  r in F.k and
A23:  for n being Nat st r in F.n holds k <= n;
A24:  (ex l being Nat st k = l + 1) implies ex s1 being Element of NAT st
      r in H.s1
      proof
        given l being Nat such that
A25:    k = l + 1;
        take k;
        reconsider l as Element of NAT by ORDINAL1:def 12;
A26:    not r in G.l
        proof
          assume r in G.l;
          then consider i being Nat such that
A27:      i <= l and
A28:      r in F.i by A1,Th5;
          k <= i by A23,A28;
          hence thesis by A25,A27,NAT_1:13;
        end;
        H.(l+1) = F.(l+1) \ G.l by A3;
        hence thesis by A22,A25,A26,XBOOLE_0:def 5;
      end;
      k=0 implies ex s1 being Element of NAT st r in H.s1 by A2,A22;
      hence thesis by A24,NAT_1:6;
    end;
    then consider s1 being Element of NAT such that
A29: r in H.s1;
    H.s1 in rng H by FUNCT_2:4;
    hence thesis by A29,TARSKI:def 4;
  end;
  assume T = rng F;
  hence thesis by A10,A6,A12,SUPINF_2:43;
end;
