reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th11:
 seq1 is convergent & seq2 is convergent & seq1 is nonnegative &
seq2 is nonnegative & (for k be Nat holds seq.k = seq1.k + seq2.k) implies seq
  is nonnegative & seq is convergent & lim seq = lim seq1 + lim seq2
proof
  assume that
A1: seq1 is convergent and
A2: seq2 is convergent and
A3: seq1 is nonnegative and
A4: seq2 is nonnegative and
A5: for k be Nat holds seq.k = seq1.k + seq2.k;
A6: not seq2 is convergent_to_-infty by A4,Th8;
  for n be object st n in dom seq holds 0. <= seq.n
  proof
    let n be object;
    assume n in dom seq;
    then reconsider n1=n as Nat;
A7: 0 <= seq2.n1 by A4,SUPINF_2:51;
A8: seq.n1 = seq1.n1 + seq2.n1 by A5;
    0 <= seq1.n1 by A3,SUPINF_2:51;
    hence thesis by A7,A8;
  end;
  hence seq is nonnegative by SUPINF_2:52;
A9: not seq1 is convergent_to_-infty by A3,Th8;
  for n be Nat holds 0 <= seq2.n by A4,SUPINF_2:51;
  then
A10: lim seq2 <> -infty by A2,Th10;
  per cases by A1,A9;
  suppose
A11: seq1 is convergent_to_+infty;
    for g be Real st 0 < g ex n be Nat st for m be Nat st n<=m
    holds g <= seq.m
    proof
      let g be Real;
      assume 0 < g;
      then consider n be Nat such that
A12:  for m be Nat st n <= m holds g <= seq1.m by A11;
      take n;
      let m be Nat;
      assume n<=m;
      then
A13:  g <= seq1.m by A12;
      0 <= seq2.m by A4,SUPINF_2:51;
      then  g + zz <= seq1.m + seq2.m by A13,XXREAL_3:36;
      then g <= seq1.m + seq2.m by XXREAL_3:4;
      hence thesis by A5;
    end;
    then
A14: seq is convergent_to_+infty;
    hence seq is convergent;
    then
A15: lim seq = +infty by A14,MESFUNC5:def 12;
    lim seq1 = +infty by A1,A11,MESFUNC5:def 12;
    hence thesis by A10,A15,XXREAL_3:def 2;
  end;
  suppose
A16: seq1 is convergent_to_finite_number;
    then
A17: not seq1 is convergent_to_-infty by MESFUNC5:51;
    not seq1 is convergent_to_+infty by A16,MESFUNC5:50;
    then consider g be Real such that
A18: lim seq1 = g and
A19: for p be Real st 0<p ex n be Nat st for m be Nat st n<=m
    holds |. seq1.m - lim seq1 .| < p and
    seq1 is convergent_to_finite_number by A1,A17,MESFUNC5:def 12;
    per cases by A2,A6;
    suppose
A20:  seq2 is convergent_to_+infty;
      for g be Real st 0 < g ex n be Nat st for m be Nat st n<=m
      holds g <= seq.m
      proof
        let g be Real;
        assume 0 < g;
        then consider n be Nat such that
A21:    for m be Nat st n <= m holds g <= seq2.m by A20;
        take n;
        let m be Nat;
        assume n<=m;
        then
A22:    g <= seq2.m by A21;
        0 <= seq1.m by A3,SUPINF_2:51;
        then  g + zz <= seq1.m + seq2.m by A22,XXREAL_3:36;
        then g <= seq1.m + seq2.m by XXREAL_3:4;
        hence thesis by A5;
      end;
      then
A23:  seq is convergent_to_+infty;
      hence seq is convergent;
      then
A24:  lim seq = +infty by A23,MESFUNC5:def 12;
      lim seq2 = +infty by A2,A20,MESFUNC5:def 12;
      hence thesis by A18,A24,XXREAL_3:def 2;
    end;
    suppose
A25:  seq2 is convergent_to_finite_number;
      then
A26:  seq2 is not convergent_to_-infty by MESFUNC5:51;
      seq2 is not convergent_to_+infty by A25,MESFUNC5:50;
      then consider h be Real such that
A27:  lim seq2 = h and
A28:  for p be Real st 0<p ex n be Nat st for m be Nat st n<=m
      holds |. seq2.m - lim seq2 .| < p and
      seq2 is convergent_to_finite_number by A2,A26,MESFUNC5:def 12;
      reconsider h9=h, g9=g as Real;
      reconsider gh = g+h as R_eal by XXREAL_0:def 1;
A29:  for p be Real st 0<p ex n be Nat st for m be Nat st n <= m
      holds |. seq.m - (g+h).| < p
      proof
       let p be Real;
        reconsider pp= p as Element of REAL by XREAL_0:def 1;
        assume
A30:    0 < p;
        then consider n1 be Nat such that
A31:    for m be Nat st n1 <= m holds |. seq1.m - lim seq1 .| < p/2 by A19;
        consider n2 be Nat such that
A32:    for m be Nat st n2 <= m holds |. seq2.m - lim seq2 .| < p/2 by A28,A30;
        reconsider n19=n1, n29=n2 as Element of NAT by ORDINAL1:def 12;
        reconsider n = max(n19,n29) as Nat;
        take n;
        let m be Nat;
        assume
A33:    n <= m;
A34:     pp/2 in REAL by XREAL_0:def 1;
        n2 <= n by XXREAL_0:25;
        then n2 <= m by A33,XXREAL_0:2;
        then
A35:    |. seq2.m - lim seq2 .| < pp/2 by A32;
        then |. seq2.m - lim seq2 .| < +infty by XXREAL_0:2,9,A34;
        then
A36:    seq2.m -  h in REAL by A27,EXTREAL1:41;
        n1 <= n by XXREAL_0:25;
        then n1 <= m by A33,XXREAL_0:2;
        then
A37:    |. seq1.m - lim seq1 .| < pp/2 by A31;
        then |. seq1.m - lim seq1 .| < +infty by XXREAL_0:2,9,A34;
        then seq1.m -  g in REAL by A18,EXTREAL1:41;
        then consider e1,e2 be Real such that
A38:    e1 = seq1.m -  g and
A39:    e2 = seq2.m -  h by A36;
A40:    |. seq2.m -  h .| = |.e2 qua Complex.| by A39,EXTREAL1:12;
A41:    0 <= seq2.m by A4,SUPINF_2:51;
        then
A42:    seq2.m -  h <> -infty by XXREAL_3:19;
A43:    0 <= seq1.m by A3,SUPINF_2:51;
A44:    |. seq1.m -  g .| = |.e1 qua Complex.| by A38,EXTREAL1:12;
        then
A45:    |. seq2.m -  h .| + |. seq1.m -  g .|
             = |.e1 qua Complex.| + |.e2 qua Complex.|
        by A40,SUPINF_2:1;
        (g+h) =  g +  (h qua ExtReal);
        then seq.m - (g+h) = seq.m -  h -  g by XXREAL_3:31
          .= seq1.m + seq2.m -  h -  g by A5
          .= seq1.m + (seq2.m -  h) -  g by A43,A41,XXREAL_3:30
          .= (seq2.m -  h) + (seq1.m -  g) by A43,A42,XXREAL_3:30;
        then
A46:    |. seq.m - (g+h) .| <= |. seq2.m -  h .| + |. seq1.m - g .|
                by EXTREAL1:24;
        |.e1 qua Complex.| + |.e2 qua Complex.| < p/2 + p/2
          by A18,A27,A37,A35,A44,A40,XREAL_1:8;
        hence thesis by A46,A45,XXREAL_0:2;
      end;
      then
A47:  seq is convergent_to_finite_number;
      hence seq is convergent;
      then lim seq = gh by A29,A47,MESFUNC5:def 12;
      hence thesis by A18,A27,SUPINF_2:1;
    end;
  end;
end;
