
theorem Th9:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
  of S, F be Finite_Sep_Sequence of S holds M.(union rng F) = Sum(M*F)
proof
  let X be non empty set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let F be Finite_Sep_Sequence of S;
  reconsider F1 = M*F as FinSequence of ExtREAL;
  consider f be sequence of ExtREAL such that
A1: Sum(F1) = f.(len F1) and
A2: f.0 = 0. and
A3: for i be Nat st i < len F1 holds f.(i+1) = f.i + F1.(i+1)
   by EXTREAL1:def 2;
  consider G being Sep_Sequence of S such that
A4: union rng F = union rng G and
A5: for n being Nat st n in dom F holds F.n = G.n and
A6: for m being Nat st not m in dom F holds G.m = {} by MESFUNC2:30;
  defpred Q[Nat] means $1 <= len F1 implies Ser(M*G).$1 = f.$1;
  set G1 = M*G;
A7: dom G = NAT by FUNCT_2:def 1;
A9: for i be Nat st i < len F1 holds Ser(M*G).(i+1) = Ser(M*G).i + F1.(i+1)
  proof
    let i be Nat;
    assume i < len F1;
    then 1 <= i+1 & i+1 <= len F1 by NAT_1:11,13;
    then i+1 in dom F1 by FINSEQ_3:25;
    then
A10: i+1 in dom F by Th8;
    reconsider i as Element of NAT by ORDINAL1:def 12;
A11: i+1 in NAT by ORDINAL1:def 12;
    Ser(M*G).(i+1) = Ser(M*G).i + (M*G).(i+1) by SUPINF_2:def 11
      .= Ser(M*G).i + M.(G.(i+1)) by A7,FUNCT_1:13,A11
      .= Ser(M*G).i + M.(F.(i+1)) by A5,A10;
    hence thesis by A10,FUNCT_1:13;
  end;
A12: for k be Nat st Q[k] holds Q[k+1]
  proof
A13: for i be Nat st i < len F1 holds f.(i+1) = f.i + F1.(i+1)
    by A3;
    let k be Nat;
    assume
A14: Q[k];
    assume k+1 <= len F1;
    then
A15: k < len F1 by NAT_1:13;
    then Ser(M*G).(k+1) = f.k + F1.(k+1) by A9,A14;
    hence thesis by A13,A15;
  end;
  defpred P[Nat] means $1 >= len F1 implies Ser(M*G).$1 = Ser(M*G).(len F1);
A16: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A17: P[k];
    assume
A18: k+1 >= len F1;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    now
      per cases;
      case
A19:    k >= len F1;
        then k+1 > len F1 by NAT_1:13;
        then not k+1 in Seg len F1 by FINSEQ_1:1;
        then not k+1 in dom F1 by FINSEQ_1:def 3;
        then
A20:    not k+1 in dom F by Th8;
        k+1 in NAT by ORDINAL1:def 12;
        then
A21:    G1.(k+1) = M.(G.(k+1)) by A7,FUNCT_1:13
          .= M.{} by A6,A20
          .= 0. by VALUED_0:def 19;
        Ser(M*G).(k+1) = Ser(M*G).(len F1) + G1.(k+1) by A17,A19,
SUPINF_2:def 11;
        hence thesis by A21,XXREAL_3:4;
      end;
      case
        k < len F1;
        then k+1 <= len F1 by NAT_1:13;
        hence thesis by A18,XXREAL_0:1;
      end;
    end;
    hence thesis;
  end;
  defpred P1[Nat] means $1 < len F1 implies Ser(M*G).(len F1 - $1) <= Ser(M*G)
  .(len F1);
A22: M*G is nonnegative by MEASURE2:1;
A23: for k be Nat st P1[k] holds P1[k+1]
  proof
    let k be Nat;
    assume
A24: P1[k];
    assume
A25: k+1 < len F1;
    then consider k9 being Nat such that
A26: len F1 = (k+1 qua Nat) + k9 by NAT_1:10;
    reconsider k9 as Element of NAT by ORDINAL1:def 12;
    k <= k+1 & Ser(M*G).(k9) <= Ser(M*G).(k9 + 1) by A22,NAT_1:11,SUPINF_2:40;
    hence thesis by A24,A25,A26,XXREAL_0:2;
  end;
  not 0 in Seg len F by FINSEQ_1:1;
  then not 0 in dom F by FINSEQ_1:def 3;
  then
A27: G.0 = {} by A6;
  Ser(M*G).0 = G1.0 by SUPINF_2:def 11;
  then
A28: Ser(M*G).0 = M.(G.0) by A7,FUNCT_1:13
    .= 0. by A27,VALUED_0:def 19;
  then
A29: Q[0] by A2;
A30: for k be Nat holds Q[k] from NAT_1:sch 2(A29,A12);
A31: P1[0];
A32: for i be Nat holds P1[i] from NAT_1:sch 2(A31,A23);
A33: for i be Nat st i < len F1 holds Ser(M*G).i <= Ser(M*G).(len F1)
  proof
    let i be Nat;
A34: len F1 <= len F1 + i by NAT_1:11;
    assume i < len F1;
    then consider k be Nat such that
A35: len F1 = i + k by NAT_1:10;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    k = len F1 - i by A35;
    then
A36: k <= len F1 by A34,XREAL_1:20;
    Ser(M*G).(len F1 - k) <= Ser(M*G).(len F1)
    proof
      now
        per cases by A36,XXREAL_0:1;
        case
          k = len F1;
          hence thesis by A28,A22,SUPINF_2:40;
        end;
        case
          k < len F1;
          hence thesis by A32;
        end;
      end;
      hence thesis;
    end;
    hence thesis by A35;
  end;
A37: P[0];
A38: for k be Nat holds P[k] from NAT_1:sch 2(A37,A16);
  for z be ExtReal st z in rng Ser(M*G) holds z <= Ser(M*G).(len F1)
  proof
    let z be ExtReal;
    assume z in rng Ser(M*G);
    then consider n be object such that
A39: n in dom Ser(M*G) and
A40: z = Ser(M*G).n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A39;
    now
      per cases;
      case
        n < len F1;
        hence thesis by A33,A40;
      end;
      case
        n >= len F1;
        hence thesis by A38,A40;
      end;
    end;
    hence thesis;
  end;
  then
A41: Ser(M*G).(len F1) is UpperBound of rng Ser(M*G) by XXREAL_2:def 1;
  dom (Ser(M*G)) = NAT by FUNCT_2:def 1;
  then
A42: Ser(M*G).(len F1) = sup(rng Ser(M*G)) by A41,FUNCT_1:3,XXREAL_2:55;
  M.(union rng F) = SUM(M*G) by A4,MEASURE1:def 6
    .= sup(rng Ser(M*G));
  hence thesis by A1,A30,A42;
end;
