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;

theorem Th3: :: extension of MEASURE7:4
  seq1 is nonnegative & seq2 is nonnegative & (for n being Nat
holds seq.n = seq1.n + seq2.n) implies seq is nonnegative & SUM seq = SUM seq1
  + SUM seq2 & Sum seq = Sum seq1 + Sum seq2
proof
  assume that
A1: seq1 is nonnegative and
A2: seq2 is nonnegative;
  reconsider Hseq = Ser seq2 as ExtREAL_sequence;
  reconsider Gseq = Ser seq1 as ExtREAL_sequence;
  reconsider Fseq = Ser seq as ExtREAL_sequence;
  assume
A3: for n being Nat holds seq.n = seq1.n + seq2.n;
  then
A4: for n be Element of NAT holds seq.n = seq1.n + seq2.n;
A5: for k be Nat holds Fseq.k = Gseq.k + Hseq.k
  by A1,A2,A4,MEASURE7:3;
  for m,n being ExtReal st m in dom Fseq & n in dom Fseq & m <= n
  holds Fseq.m <= Fseq.n
  proof
    let m,n be ExtReal;
    assume that
A6: m in dom Fseq and
A7: n in dom Fseq and
A8: m <= n;
    Gseq.m <= Gseq.n & Hseq.m <= Hseq.n by A1,A2,A6,A7,A8,MEASURE7:8;
    then Gseq.m + Hseq.m <= Gseq.n + Hseq.n by XXREAL_3:36;
    then Fseq.m <= Gseq.n + Hseq.n by A1,A2,A4,A6,MEASURE7:3;
    hence thesis by A1,A2,A4,A7,MEASURE7:3;
  end;
  then Fseq is non-decreasing by VALUED_0:def 15;
  then
A9: lim Fseq = sup Fseq by RINFSUP2:37;
  Partial_Sums seq1 is non-decreasing by A1,MESFUNC9:16;
  then Gseq is non-decreasing by Th1;
  then
A10: Gseq is convergent & lim Gseq = sup Gseq by RINFSUP2:37;
  Partial_Sums seq2 is nonnegative by A2,MESFUNC9:16;
  then
A11: Hseq is nonnegative by Th1;
  now
    let n be object;
    assume n in dom seq;
    then reconsider n9=n as Element of NAT;
A12: seq2.n9 >= 0 by A2,SUPINF_2:51;
    seq.n = seq1.n9 + seq2.n9 & seq1.n9 >= 0 by A1,A3,SUPINF_2:51;
    hence seq.n >= 0 by A12;
  end;
  hence
A13: seq is nonnegative by SUPINF_2:52;
  Partial_Sums seq2 is non-decreasing by A2,MESFUNC9:16;
  then Hseq is non-decreasing by Th1;
  then
A14: Hseq is convergent & lim Hseq = sup Hseq by RINFSUP2:37;
  Partial_Sums seq1 is nonnegative by A1,MESFUNC9:16;
  then Gseq is nonnegative by Th1;
  hence SUM seq = SUM seq1 + SUM seq2 by A10,A11,A14,A5,A9,MESFUNC9:11;
  then Sum seq = SUM seq1 + SUM seq2 by A13,Th2;
  then Sum seq = Sum seq1 + SUM seq2 by A1,Th2;
  hence Sum seq = Sum seq1 + Sum seq2 by A2,Th2;
end;
