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;
reserve C for C_Measure of X;

theorem Th26:
  for X being non empty set, S be SigmaField of X, M be
  sigma_Measure of S, SSets being SetSequence of S st SSets is non-descending
  holds lim(M*SSets) = M.(lim SSets)
proof
  let X be non empty set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let SSets be SetSequence of S;
  assume
A1: SSets is non-descending;
  then
A2: Partial_Union SSets = SSets by PROB_4:19;
  rng Partial_Diff_Union SSets c= S;
  then reconsider Bseq = Partial_Diff_Union SSets as Sep_Sequence of S
      by FUNCT_2:6;
  for n be object st n in NAT holds Ser(M*Bseq).n = (M*Partial_Union SSets) .n
  proof
    defpred P[Nat] means Ser(M*Bseq).$1 = (M*Partial_Union SSets).$1;
    let n be object;
A3: for k being Nat st P[k] holds P[k + 1]
    proof
      let k be Nat;
      assume
A4:   P[k];
A5:   (Partial_Union (Partial_Diff_Union SSets)).k = (Partial_Union
SSets).k & (Partial_Diff_Union SSets).(k+1) = SSets.(k+1) \ (Partial_Union
      SSets).k by PROB_3:35,def 3;
A6:   k in NAT by ORDINAL1:def 12;
      Ser(M*Bseq).(k+1) = Ser(M*Bseq).k + (M*Bseq).(k+1) by SUPINF_2:def 11
        .= (M*Partial_Union SSets).k + M.((Partial_Diff_Union SSets).(k+1)
      ) by A4,FUNCT_2:15
        .= M.((Partial_Union SSets).k) + M.((Partial_Diff_Union SSets).(k+
      1)) by FUNCT_2:15,A6
        .= M.((Partial_Union (Partial_Diff_Union SSets)).k) + M.((
      Partial_Diff_Union SSets).(k+1)) by PROB_3:35;
      then
      Ser(M*Bseq).(k+1) = M.( (Partial_Union (Partial_Diff_Union SSets))
      .k \/ (Partial_Diff_Union SSets).(k+1) ) by A5,MEASURE1:30,XBOOLE_1:79
        .= M.( (Partial_Union (Partial_Diff_Union SSets)).(k+1)) by
PROB_3:def 2
        .= M.( (Partial_Union SSets).(k+1)) by PROB_3:35;
      hence Ser(M*Bseq).(k+1) = (M*(Partial_Union SSets)).(k+1) by FUNCT_2:15;
    end;
    assume n in NAT;
    then reconsider n1 = n as Element of NAT;
    Ser(M*Bseq).0 = (M*Bseq).0 by SUPINF_2:def 11
      .= M.((Partial_Diff_Union SSets).0) by FUNCT_2:15
      .= M.(SSets.0) by PROB_3:31
      .= M.((Partial_Union SSets).0) by PROB_3:22;
    then
A7: P[0] by FUNCT_2:15;
    for k being Nat holds P[k] from NAT_1:sch 2(A7,A3);
    then Ser(M*Bseq).n1 = (M*Partial_Union SSets).n1;
    hence thesis;
  end;
  then
A8: Ser(M * Bseq) = M * SSets by A2,FUNCT_2:12;
  reconsider Gseq = Ser(M * Bseq) as ExtREAL_sequence;
  M*Bseq is nonnegative by MEASURE1:25;
  then
  for m,n being ExtReal st m in dom Gseq & n in dom Gseq & m <= n
  holds Gseq.m <= Gseq.n by MEASURE7:8;
  then Gseq is non-decreasing by VALUED_0:def 15;
  then
A9: SUM(M * Bseq) = M.(union rng Bseq) & lim Gseq = sup Gseq by MEASURE1:def 6
,RINFSUP2:37;
  Partial_Union (Partial_Diff_Union SSets) = SSets by A2,PROB_3:35;
  then Union SSets = Union Partial_Diff_Union SSets by PROB_3:30;
  then lim Gseq = M.(Union SSets) by A9,CARD_3:def 4;
  hence thesis by A1,A8,SETLIM_1:63;
end;
