reserve s1,s2,q1 for Real_Sequence;
reserve n for Element of NAT;
reserve a,b for Real;

theorem
for X be RealBanachSpace, f be continuous PartFunc of REAL,the carrier of X
  st a <= b & [' a,b '] c= dom f holds f is_integrable_on [' a,b ']
proof
   let X be RealBanachSpace,
       f be continuous PartFunc of REAL,the carrier of X;
   set A = [' a,b '];
   assume A1: a <= b & [' a,b '] c= dom f; then
   reconsider h=f|A as Function of A,the carrier of X by Lm1;
A2: f|A is uniformly_continuous by A1,Th3;
   consider T0 being DivSequence of A such that
A3: delta T0 is convergent & lim  delta T0=0 by INTEGRA4:11;
   set S0 = the middle_volume_Sequence of h,T0;
   set I = lim middle_sum(h,S0);
   for T being DivSequence of A,
       S be middle_volume_Sequence of h,T
    st delta T is convergent & lim delta T = 0
   holds middle_sum(h,S) is convergent & lim middle_sum(h,S) = I
   proof
    let T be DivSequence of A, S be middle_volume_Sequence of h,T;
    assume A4: delta T is convergent & lim delta T = 0;
    hence middle_sum(h,S) is convergent by A2,Th13;
    consider T1 be DivSequence of A such that
A5:  for i be Nat holds
      T1.(2*i) = T0.i & T1.(2*i+1) = T.i by Th15;
    consider S1 be middle_volume_Sequence of h,T1 such that
A6:  for i be Nat holds S1.(2*i) = S0.i & S1.(2*i+1) = S.i by A5,Th17;
    delta T1 is convergent & lim delta T1 = 0 by A4,A5,A3,Th16; then
A7: middle_sum(h,S1) is convergent by A2,Th13;
A8: for i be Nat holds
      (middle_sum(h,S1)).(2*i) = (middle_sum(h,S0)).i
    & (middle_sum(h,S1)).(2*i+1) = (middle_sum(h,S)).i
    proof
     let i be Nat;
     reconsider S1 as middle_volume_Sequence of h,T1;
     (middle_sum(h,S1)).(2*i) = middle_sum(h,S1.(2*i))
   & (middle_sum(h,S1)).(2*i+1) = middle_sum(h,S1.(2*i+1))
         by INTEGR18:def 4; then
     (middle_sum(h,S1)).(2*i) = middle_sum(h,S0.i)
   & (middle_sum(h,S1)).(2*i+1) = middle_sum(h,S.i) by A6;
     hence thesis by INTEGR18:def 4;
    end;
    lim middle_sum(h,S) = lim middle_sum(h,S1) by A7,A8,Th18;
    hence lim middle_sum(h,S) = lim middle_sum(h,S0) by A7,A8,Th18;
   end;
   then h is integrable;
   hence thesis;
end;
