reserve a,b,c,d,e for Real;
reserve X,Y for set,
          Z for non empty set,
          r for Real,
          s for ExtReal,
          A for Subset of REAL,
          f for real-valued Function;
reserve I for non empty closed_interval Subset of REAL,
       TD for tagged_division of I,
        D for Division of I,
        T for Element of set_of_tagged_Division(D),
        f for PartFunc of I,REAL;
reserve f for Function of I,REAL;

theorem Th28:
  A misses I & f = chi(A,I) implies tagged_sum(f,TD) = 0
  proof
    assume that
A1: A misses I and
A2: f = chi(A,I);
A3: dom tagged_volume(f,TD) = Seg len tagged_volume(f,TD) by FINSEQ_1:def 3
                           .= Seg len TD by Def4
                           .= dom TD by FINSEQ_1:def 3;
    for i be Nat st i in dom TD holds tagged_volume(f,TD).i = 0
    proof
      let i be Nat;
      assume
A4:   i in dom TD;
      consider D be Division of I,
               T be Element of set_of_tagged_Division(D) such that
A5:   tagged_of TD = T and
A6:   TD = [D,T] by Def2;
A7:   i in dom D by A4,Th20,A6;
A8:   dom T = Seg len tagged_of TD by A5,FINSEQ_1:def 3
           .= Seg len division_of TD by Th21
           .= Seg len D by A6,Th20
           .= dom D by FINSEQ_1:def 3;
      rng T c= I by Th22;
      then T.i in I by A8,A7,FUNCT_1:3;
      then f.((tagged_of TD).i) = 0 by A1,A5,A2,Th15;
      then tagged_volume(f,TD).i = 0 * vol(divset(division_of TD,i)) by Def4,A4
                                .= 0;
      hence thesis;
    end;
    then for k be object st k in dom tagged_volume(f,TD) holds
      (tagged_volume(f,TD)).k = 0 by A3;
    then Sum tagged_volume(f,TD) = Sum ((len tagged_volume(f,TD)) |-> 0)
                                     by INTEGR23:5
                                .= 0 by RVSUM_1:81;
    hence thesis;
  end;
