reserve X for RealNormSpace;

theorem Th17:
  for f being PartFunc of REAL,the carrier of X,
      A being non empty closed_interval Subset of REAL
        st vol(A) = 0 & A c= dom f holds
        f is_integrable_on A & integral(f,A) = 0.X
proof
  let f being PartFunc of REAL,the carrier of X,
      A being non empty closed_interval Subset of REAL;
  assume A1: vol(A) = 0 & A c= dom f;
  f is Function of dom f, rng f by FUNCT_2:1; then
  f is Function of dom f,the carrier of X by FUNCT_2:2; then
  reconsider g = f|A as Function of A,the carrier of X by A1,FUNCT_2:32;
A2: for T being DivSequence of A,
    S be middle_volume_Sequence of g,T st
    delta(T) is convergent & lim delta(T) = 0 holds
    middle_sum(g,S) is convergent & lim (middle_sum(g,S)) = 0.X
    proof
      let T being DivSequence of A,
          S be middle_volume_Sequence of g,T;
      assume delta(T) is convergent & lim delta(T) = 0;
A3:  for i be Nat holds middle_sum(g,S.i) = 0.X
      proof
        let i be Nat;
A4:     len (S.i) = len (S.i);
        now let k be Nat, v be Element of X;
          assume A5: k in dom (S.i) & v = (S.i).k; then
          k in Seg (len (S.i)) by FINSEQ_1:def 3; then
          k in Seg (len (T.i)) by Def1; then
A6:      k in dom (T.i) by FINSEQ_1:def 3; then
          consider c be VECTOR of X such that
A7:        c in rng (g|divset((T.i),k)) &
            (S.i).k = (vol divset((T.i),k)) * c by Def1;
          0 <= vol divset((T.i),k) & vol divset((T.i),k) <= 0
            by A1,A6,INTEGRA1:8,9,INTEGRA2:38; then
A8:      vol divset((T.i),k) = 0;
          (S.i).k = 0.X by A8,A7,RLVECT_1:10;
          hence (S.i).k = -v by A5,RLVECT_1:12;
        end;
        then Sum((S.i)) = -Sum((S.i)) by A4,RLVECT_1:40;
        hence thesis by RLVECT_1:19;
      end;
A9:  for i be Nat holds (middle_sum(g,S)).i = 0.X
      proof
        let i be Nat;
        thus (middle_sum(g,S)).i = middle_sum(g,S.i) by Def4
                                .= 0.X by A3;
      end;
A10:  for r be Real st 0 < r
     ex m be Nat st for n be Nat
        st m <= n holds ||.((middle_sum(g,S)).n) - 0.X.|| < r
      proof
        let r be Real;
        assume A11: 0 < r;
        take m = 0;
        let n be Nat;
        assume m <= n;
        ||.((middle_sum(g,S)).n) - 0.X.|| = ||.0.X - 0.X.|| by A9
                                         .= 0 by NORMSP_1:6;
        hence ||.((middle_sum(g,S)).n) - 0.X.|| < r by A11;
      end;
    hence (middle_sum(g,S)) is convergent by NORMSP_1:def 6;
    hence lim (middle_sum(g,S)) = 0.X by A10,NORMSP_1:def 7;
  end; then
A12: g is integrable;
  hence f is_integrable_on A;
  integral(g) = 0.X by A12,A2,Def6;
  hence integral(f,A) = 0.X by Def8,A1;
end;
