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

theorem Th13:
for X be RealBanachSpace, A be non empty closed_interval Subset of REAL,
    h be Function of A,the carrier of X
  st
   for r be Real st 0<r ex s be Real st 0<s &
     for x1,x2 be Real st x1 in dom h & x2 in dom h & |. x1-x2 .| < s
       holds ||. h/.x1-h/.x2 .|| < r
holds
   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
proof
   let X be RealBanachSpace, A be non empty closed_interval Subset of REAL,
       h be Function of A,the carrier of X;
   assume
A1: for r be Real st 0<r ex s be Real st 0<s &
     for x1,x2 be Real st x1 in dom h & x2 in dom h & |. x1-x2 .| < s
       holds ||. h/.x1-h/.x2 .|| < r;

   thus
    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
   proof
    let T being DivSequence of A, S be middle_volume_Sequence of h,T;
    assume A2:delta T is convergent & lim delta T = 0;

    defpred P[Element of NAT, set] means
     ex p be FinSequence of REAL
      st p = $2 & len p = len (T.$1)
       & for i be Nat st i in dom (T.$1)
          holds (p.i) in dom (h|divset((T.$1),i))
              & ex z be Point of X
                 st z = (h|divset((T.$1),i)).(p.i)
                 & (S.$1).i = (vol divset((T.$1),i)) * z;

A3: for k be Element of NAT ex p be Element of (REAL)* st P[k, p]
    proof
     let k be Element of NAT;
     defpred P1[Nat,set] means $2 in dom (h|divset((T.k),$1)) &
      ex c be Point of X st c = (h|divset((T.k),$1)).$2 &
        (S.k).$1 = (vol divset((T.k),$1)) * c;

A4:  Seg len(T.k) = dom(T.k) by FINSEQ_1:def 3;

A5:  for i be Nat st i in Seg len (T.k) ex x be Element of REAL st P1[i,x]
     proof
      let i be Nat;
      assume i in Seg len(T.k); then
      i in dom (T.k) by FINSEQ_1:def 3; then
      consider c be Point of X such that
A6:    c in rng (h|divset((T.k),i)) &
       (S.k).i = vol divset(T.k,i)*c by INTEGR18:def 1;

      consider x be object such that
A7:    x in dom (h|divset(T.k,i)) &
       c = (h|divset(T.k,i)).x by A6,FUNCT_1:def 3;
      x in dom h & x in divset(T.k,i) by A7,RELAT_1:57; then
      reconsider x as Element of REAL;
      take x;
      thus thesis by A6,A7;
     end;
     consider p be FinSequence of REAL such that
A8:  dom p = Seg len(T.k)
    & for i be Nat st i in Seg len(T.k) holds P1[i,p.i]
        from FINSEQ_1:sch 5(A5);

     take p;
     len p = len (T.k) by A8,FINSEQ_1:def 3;
     hence thesis by A8,A4,FINSEQ_1:def 11;
    end;
    consider Fn be sequence of (REAL)* such that
A9: for x be Element of NAT holds P[x, Fn.x] from FUNCT_2:sch 3(A3);
    consider Fm be sequence of (REAL)* such that
A10: for x be Element of NAT holds P[x, Fm.x] from FUNCT_2:sch 3(A3);

A11:0<= vol A by INTEGRA1:9;

    now let p be Real;
     set pp2= p/2;
     set pv = pp2 / (vol A + 1);
     assume p > 0; then
A12:0 < pp2 & pp2 < p by XREAL_1:215,216; then
A13:0 < pv by A11,XREAL_1:139; then
     pv*(vol A) < pv *(vol A + 1) by XREAL_1:29,68; then
     pv*(vol A) < pp2 by A11,XCMPLX_1:87; then
A14:pv*(vol A) < p by A12,XXREAL_0:2;
     set p2v = pv/2;
     consider sk be Real such that
A15:  0 < sk
    & for x1,x2 be Real st x1 in dom h & x2 in dom h & |. x1-x2 .| < sk
        holds ||. h/.x1-h/.x2 .|| < p2v by A1,A13,XREAL_1:215;
     consider k be Nat such that
A16:   for i be Nat st k <= i holds |. (delta T).i - 0 .| < sk
        by A15,A2,SEQ_2:def 7;
     take k;

     let n, m be Nat;
A17:  n in NAT & m in NAT by ORDINAL1:def 12;
     assume n >= k & m >= k; then
     |. (delta T).n - 0 .| < sk & |. (delta T).m - 0 .| < sk by A16; then
     |. delta(T.n) .| < sk & |. delta(T.m) .| < sk by INTEGRA3:def 2,A17; then
A18: delta(T.n) < sk & delta(T.m) < sk by INTEGRA3:9,ABSVALUE:def 1;

A19:  middle_sum(h,S).n = middle_sum(h,S.n)
   & middle_sum(h,S).m = middle_sum(h,S.m) by INTEGR18:def 4;

     consider p1 be FinSequence of REAL such that
A20:   p1 = Fn.n & len p1 = len (T.n)
    & for i be Nat st i in dom (T.n)
       holds p1.i in dom (h|divset(T.n,i))
           & ex z be Point of X st z = (h|divset(T.n,i)).(p1.i)
           & (S.n).i = (vol divset(T.n,i)) * z by A9,A17;
     consider p2 be FinSequence of REAL such that
A21:   p2 = Fm.m & len p2 = len (T.m)
    & for i be Nat st i in dom (T.m)
       holds p2.i in dom (h|divset(T.m,i))
           & ex z be Point of X st z = (h|divset(T.m,i)).(p2.i)
           & (S.m).i = (vol divset(T.m,i)) * z by A10,A17;
     defpred H1[object,object,object] means
       ex i,j being Nat, z be Point of X
        st $1=i & $2=j & z = (h|divset(T.n,i)).(p1.i)
         & $3 = (xvol (divset(T.n,i) /\ divset(T.m,j)))* z;

A22: for x,y be object st x in Seg len(T.n) & y in Seg len (T.m)
       ex w be object st w in the carrier of X & H1[x,y,w]
     proof
      let x,y be object;
      assume A23: x in Seg len (T.n) & y in Seg len (T.m);
      then reconsider i=x,j=y as Nat;

      i in dom (T.n) by FINSEQ_1:def 3,A23;
      then consider z be Point of X such that
A24:   z = (h|divset(T.n,i)).(p1.i)
     & (S.n).i = (vol divset(T.n,i)) * z by A20;

      (xvol (divset(T.n,i) /\ divset(T.m,j)))* z in the carrier of X;
      hence thesis by A24;
     end;
     consider Snm being Function of
        [: Seg len (T.n),Seg len (T.m) :],the carrier of X such that
A25:  for x,y be object st x in Seg len(T.n) & y in Seg len(T.m)
         holds H1[x,y,Snm.(x,y)] from BINOP_1:sch 1(A22);

A26:  for i,j being Nat st i in Seg len (T.n) & j in Seg len (T.m)
      holds
       ex z be Point of X
         st z = (h|divset(T.n,i)).(p1.i)
       & Snm.(i,j) = (xvol (divset(T.n,i) /\ divset(T.m,j)))* z
     proof
      let i,j being Nat;
      assume i in Seg len (T.n) & j in Seg len (T.m);
      then ex i1,j1 being Nat, z be Point of X
        st i=i1 & j=j1 & z = (h|divset(T.n,i1)).(p1.i1)
         & Snm.(i,j)= (xvol (divset(T.n,i1) /\ divset(T.m,j1)))* z by A25;
      hence thesis;
     end;

     defpred P1[Nat,object] means
       ex r be FinSequence of X st
         dom r = Seg len (T.m) & $2=Sum r &
         for j be Nat st j in dom r holds r.j=Snm.($1,j);

A27: for k be Nat st k in Seg len (T.n) ex x be object st P1[k,x]
     proof
      let k be Nat;
      assume A28: k in Seg len (T.n);
      deffunc G(set)= Snm.( k,$1 );

      consider r being FinSequence such that
A29:   len r = len (T.m) and
A30:   for j be Nat st j in dom r holds r.j=G(j) from FINSEQ_1:sch 2;

A31:  dom r = Seg len (T.m) by A29,FINSEQ_1:def 3;

      for j be Nat st j in dom r holds r.j in the carrier of X
      proof
       let j be Nat;
       assume A32: j in dom r; then
       [k,j] in [: Seg len (T.n), Seg len (T.m) :]
                 by A31,A28,ZFMISC_1:87; then
       Snm.(k,j) in the carrier of X by FUNCT_2:5;
       hence thesis by A30,A32;
      end;
      then reconsider r as FinSequence of X by FINSEQ_2:12;
      take x=Sum r;
      thus thesis by A30,A31;
     end;
     consider Xp be FinSequence such that
A33:  dom Xp = Seg len (T.n)
    & for k be Nat st k in Seg len (T.n) holds P1[k,Xp.k]
        from FINSEQ_1:sch 1(A27);

     for i be Nat st i in dom Xp holds Xp.i in the carrier of X
     proof
      let i be Nat;
      assume i in dom Xp;
      then ex r be FinSequence of X st dom r = Seg len (T.m) & Xp.i=Sum r
               & for j be Nat st j in dom r holds r.j=Snm.(i,j) by A33;
      hence thesis;
     end;
     then reconsider Xp as FinSequence of X by FINSEQ_2:12;

A34: len Xp = len (T.n) by FINSEQ_1:def 3,A33;

     for k be Nat st 1 <= k & k <= len Xp holds Xp.k = (S.n).k
     proof
      let k be Nat;
      assume 1 <= k & k <= len Xp; then
A35: k in Seg len Xp & k in Seg len (T.n) by A34; then
A36:  k in dom Xp & k in dom (T.n) by FINSEQ_1:def 3; then
      consider z be Point of X such that
A37:   z = (h|divset(T.n,k)).(p1.k)
     & (S.n).k = (vol divset(T.n,k)) * z by A20;
      consider r be FinSequence of X such that
A38:   dom r = Seg len (T.m) & Xp.k = Sum r
     & for j be Nat st j in dom r holds r.j=Snm.(k,j) by A35,A33;

      defpred P11[Nat,set] means $2 = xvol (divset(T.n,k) /\ divset(T.m,$1));

A39: for i be Nat st i in Seg len r holds
         ex x be Element of REAL st P11[i,x]
      proof
       let i be Nat;
       assume i in Seg len r;
       xvol (divset(T.n,k) /\ divset(T.m,i)) in REAL by XREAL_0:def 1;
       hence thesis;
      end;
      consider vtntm be FinSequence of REAL such that
A40:  dom vtntm = Seg len r & for i be Nat st i in Seg len r
         holds P11[i,vtntm.i] from FINSEQ_1:sch 5(A39);

A41:  dom vtntm = dom r
    & for j be Nat st j in dom vtntm holds
        vtntm.j=xvol (divset(T.n,k) /\ divset(T.m,j)) by A40,FINSEQ_1:def 3;

A42:  len vtntm = len r & len (T.m) = len r by A38,A40,FINSEQ_1:def 3; then
A43:  Sum vtntm = vol (divset(T.n,k)) by Th8,A40,A36,INTEGRA1:8;

      for j be Nat st j in dom r holds
        ex x be Real st x = vtntm.j & r.j = x*z
      proof
       let j be Nat;
       assume A44: j in dom r; then
A45:   ex w be Point of X
        st w = (h|divset(T.n,k)).(p1.k)
         & Snm.(k,j) = (xvol (divset(T.n,k) /\ divset(T.m,j))) * w
             by A26,A35,A38;
       take vtntm.j;
       r.j= (xvol (divset(T.n,k) /\ divset(T.m,j))) * z by A37,A45,A44,A38;
       hence thesis by A41,A44;
      end;
      hence thesis by A37,A38,A43,Th7,A42;
     end; then
A46:Xp = S.n by INTEGR18:def 1,A34;

     defpred P2[Nat,object] means
       ex s be FinSequence of X st
         dom s = Seg len (T.n) & $2=Sum s &
          for i be Nat st i in dom s holds s.i=Snm.( i,$1 );

A47:for k be Nat st k in Seg len (T.m) ex x be object st P2[k,x]
     proof
      let k be Nat;
      assume A48: k in Seg len (T.m);
      deffunc G(set)= Snm.($1,k);
      consider s being FinSequence such that
A49:  len s = len (T.n) and
A50:  for i be Nat st i in dom s holds s.i=G(i) from FINSEQ_1:sch 2;

A51: dom s = Seg len (T.n) by A49,FINSEQ_1:def 3;

      for i be Nat st i in dom s holds s.i in the carrier of X
      proof
       let i be Nat;
       assume A52: i in dom s; then
       [i,k] in [: Seg len (T.n), Seg len (T.m) :]
         by A51,A48,ZFMISC_1:87; then
       Snm.(i,k) in the carrier of X by FUNCT_2:5;
       hence thesis by A50,A52;
      end;
      then reconsider s as FinSequence of X by FINSEQ_2:12;

      take x = Sum s;
      thus thesis by A50,A51;
     end;
     consider Xq be FinSequence such that
A53:  dom Xq = Seg len (T.m)
    & for k be Nat st k in Seg len (T.m) holds P2[k,Xq.k]
             from FINSEQ_1:sch 1(A47);

     for j be Nat st j in dom Xq holds Xq.j in the carrier of X
     proof
      let j be Nat;
      assume j in dom Xq;
      then ex s be FinSequence of X st dom s = Seg len (T.n)
             & Xq.j=Sum s
             & for i be Nat st i in dom s holds s.i=Snm.(i,j) by A53;
      hence thesis;
     end;
     then reconsider Xq as FinSequence of X by FINSEQ_2:12;

     defpred H2[object,object,object] means
        ex i,j being Nat, z be Point of X
         st $1=i & $2=j & z = (h|divset(T.m,j)).(p2.j)
          & $3 = (xvol (divset(T.n,i) /\ divset(T.m,j))) * z;

A54:for x,y be object st x in Seg len (T.n) & y in Seg len (T.m)
       ex w be object st w in the carrier of X & H2[x,y,w]
     proof
      let x,y be object;
      assume A55: x in Seg len (T.n) & y in Seg len (T.m);
      then reconsider i=x,j=y as Nat;
      j in dom (T.m) by FINSEQ_1:def 3,A55;
      then consider z be Point of X such that
A56:   z = (h|divset(T.m,j)).(p2.j)
     & (S.m).j = (vol divset(T.m,j)) * z by A21;

      (xvol (divset(T.n,i) /\ divset(T.m,j)))* z in the carrier of X;
      hence thesis by A56;
     end;
     consider Smn being Function of
        [: Seg len (T.n),Seg len (T.m) :],the carrier of X such that
A57: for x,y be object st x in Seg len (T.n) & y in Seg len (T.m)
        holds H2[x,y,Smn.(x,y)] from BINOP_1:sch 1(A54);

A58: for i,j being Nat st i in Seg len (T.n) & j in Seg len (T.m)
      holds
       ex z be Point of X st z = (h|divset(T.m,j)).(p2.j)
        & Smn.(i,j) = (xvol (divset(T.n,i) /\ divset(T.m,j)))* z
     proof
      let i,j being Nat;
      assume i in Seg len (T.n) & j in Seg len (T.m);
      then ex i1,j1 being Nat, z be Point of X
         st i=i1 & j=j1 & z = (h|divset(T.m,j1)).(p2.j1)
          & Smn.(i,j)= (xvol (divset(T.n,i1) /\ divset(T.m,j1))) * z by A57;
      hence thesis;
     end;

     defpred P3[Nat,object] means
      ex s be FinSequence of X st
       dom s = Seg len (T.n) & $2=Sum s &
       for i be Nat st i in dom s holds s.i=Smn.(i,$1);

A59:for k be Nat st k in Seg len (T.m) ex x be object st P3[k,x]
     proof
      let k be Nat;
      assume A60: k in Seg len (T.m);
      deffunc G(set)= Smn.($1,k);
      consider s being FinSequence such that
A61:  len s = len (T.n) and
A62:  for i be Nat st i in dom s holds s.i=G(i) from FINSEQ_1:sch 2;

A63: dom s = Seg len (T.n) by A61,FINSEQ_1:def 3;

      for i be Nat st i in dom s holds s.i in the carrier of X
      proof
       let i be Nat;
       assume A64: i in dom s; then
       [i,k] in [: Seg len (T.n), Seg len (T.m) :]
                 by A63,A60,ZFMISC_1:87; then
       Smn.(i,k) in the carrier of X by FUNCT_2:5;
       hence thesis by A62,A64;
      end;
      then reconsider s as FinSequence of X by FINSEQ_2:12;
      take x = Sum s;
      thus thesis by A62,A63;
     end;
     consider Zq be FinSequence such that
A65: dom Zq = Seg len (T.m)
    & for k be Nat st k in Seg len (T.m) holds P3[k,Zq.k]
        from FINSEQ_1:sch 1(A59);

     for j be Nat st j in dom Zq holds Zq.j in the carrier of X
     proof
      let j be Nat;
      assume j in dom Zq;
      then ex s be FinSequence of X st dom s = Seg len (T.n)
               & Zq.j=Sum s
               & for i be Nat st i in dom s holds s.i=Smn.( i,j ) by A65;
      hence thesis;
     end;
     then reconsider Zq as FinSequence of X by FINSEQ_2:12;
A66: len Zq = len (T.m) by FINSEQ_1:def 3,A65;
     for k be Nat st 1 <= k & k <= len Zq holds Zq.k = (S.m).k
     proof
      let k be Nat;
      assume A67: 1 <= k & k <= len Zq; then
      consider s be FinSequence of X such that
A68:   dom s = Seg len (T.n) & Zq.k = Sum s
     & for i be Nat st i in dom s holds s.i=Smn.(i,k) by A65,FINSEQ_3:25;

A69:  k in Seg len (T.m) by A67,A66;

A70:  k in dom (T.m) by A67,A66,FINSEQ_3:25; then
      consider z be Point of X such that
A71:   z = (h|divset((T.m),k)).(p2.k)
     & (S.m).k = (vol divset((T.m),k)) * z by A21;
      defpred P11[Nat,set] means
       $2 = xvol (divset(T.n,$1) /\ divset(T.m,k));
A72: for i be Nat st i in Seg len s holds
        ex x be Element of REAL st P11[i,x]
      proof
       let i be Nat;
       assume i in Seg len s;
       xvol (divset(T.n,i) /\ divset(T.m,k)) in REAL by XREAL_0:def 1;
       hence thesis;
      end;
      consider vtntm be FinSequence of REAL such that
A73:  dom vtntm = Seg len s & for i be Nat st i in Seg len s
        holds P11[i,vtntm.i] from FINSEQ_1:sch 5(A72);
A74:  dom vtntm = dom s & len vtntm = len s by A73,FINSEQ_1:def 3;
A75:  for j be Nat st j in dom vtntm holds
          vtntm.j=xvol (divset(T.m,k) /\ divset(T.n,j)) by A73;
      len s = len (T.n) by A68,FINSEQ_1:def 3; then
A76:  Sum vtntm = vol divset(T.m,k) by Th8,A75,A74,A70,INTEGRA1:8;
      for j be Nat st j in dom s holds
       ex x be Real st x= vtntm.j & s.j=x * z
      proof
       let j be Nat;
       assume A77: j in dom s; then
A78:   ex w be Point of X
         st w = (h|divset((T.m),k)).(p2.k)
          & Smn.(j,k) = (xvol (divset(T.n,j) /\ divset(T.m,k))) * w
              by A58,A69,A68;
       take vtntm.j;
       s.j= (xvol (divset(T.n,j) /\ divset(T.m,k)))* z by A71,A78,A77,A68;
       hence thesis by A77,A73,A74;
      end;
      hence thesis by A71,A68,A76,Th7,A74;
     end; then
     Zq = S.m by INTEGR18:def 1,A66; then
A79: Sum(S.n) - Sum(S.m) = Sum Xq - Sum Zq by Th4,A33,A53,A46;
     set XZq = Xq - Zq;
A80: dom XZq = dom Xq /\ dom Zq by VFUNCT_1:def 2; then
     reconsider XZq = Xq - Zq as FinSequence by A53,A65,FINSEQ_1:def 2;
     now let i be Nat;
      assume i in dom XZq; then
      XZq.i = (Xq - Zq)/.i by PARTFUN1:def 6;
      hence XZq.i in the carrier of X;
     end; then
     reconsider XZq = Xq - Zq as FinSequence of X by FINSEQ_2:12;
     len Xq = len Zq by FINSEQ_3:29,A53,A65; then
A81: Sum(S.n) - Sum(S.m) = Sum (XZq) by A79,INTEGR18:2;
A82:  for i,j be Nat, Snmij,Smnij be Point of X
      st i in Seg len (T.n) & j in Seg len (T.m) & Snmij = Snm.(i,j)
       & Smnij = Smn.(i,j) holds
         ||. Snmij - Smnij .|| <= (xvol (divset(T.n,i) /\ divset(T.m,j))) * pv
     proof
      let i,j be Nat, Snmij,Smnij be Point of X;
      assume A83: i in Seg len (T.n) & j in Seg len (T.m)
                & Snmij = Snm.(i,j) & Smnij = Smn.(i,j); then
      consider z1 be Point of X such that
A84:   z1 = (h|divset(T.n,i)).(p1.i)
     & Snm.(i,j) = (xvol (divset(T.n,i) /\ divset(T.m,j)))* z1 by A26;
      consider z2 be Point of X such that
A85:   z2 = (h|divset((T.m),j)).(p2.j)
     & Smn.(i,j) = (xvol (divset(T.n,i) /\ divset(T.m,j))) * z2 by A58,A83;

A86:  i in dom (T.n) & j in dom (T.m) by A83,FINSEQ_1:def 3; then
A87:  p1.i in dom (h|divset(T.n,i))
    & p2.j in dom (h|divset(T.m,j)) by A20,A21; then
      p1.i in dom h /\ divset(T.n,i)
    & p2.j in dom h /\ divset(T.m,j) by RELAT_1:61; then
A88:  p1.i in dom h & p1.i in divset(T.n,i)
    & p2.j in dom h & p2.j in divset(T.m,j) by XBOOLE_0:def 4;

      z1 = h.(p1.i) & z2 = h.(p2.j) by A84,A85,A87,FUNCT_1:47; then
A89:  z1 = h/.(p1.i) & z2 =h/.(p2.j) by A88,PARTFUN1:def 6;

      per cases;
      suppose A90:divset(T.n,i) /\ divset(T.m,j) = {}; then
A91:    xvol (divset(T.n,i) /\ divset(T.m,j)) = (0 qua Real) by Def2;
       Snmij =0.X & Smnij = 0.X by A83,A84,A85,A90,Def2,RLVECT_1:10;
       hence ||. Snmij - Smnij .||
         <= (xvol (divset(T.n,i) /\ divset(T.m,j))) * pv by A91;
      end;
      suppose divset(T.n,i) /\ divset(T.m,j) <> {}; then
       consider t be object such that
A92:   t in divset(T.n,i) /\ divset(T.m,j) by XBOOLE_0:def 1;
       reconsider t as Real by A92;
A93:  dom h = A by FUNCT_2:def 1;
A94:  divset(T.m,j) c= A by A86,INTEGRA1:8;
A95:   t in divset(T.n,i) & t in divset(T.m,j) by A92,XBOOLE_0:def 4; then
       |. (p1.i)-t .| < sk & |. t-(p2.j) .| < sk by A86,A88,Th12,A18; then
A96:   ||. h/.(p1.i)-h/.t .|| < p2v
     & ||. h/.t-h/.(p2.j) .|| < p2v by A94,A95,A93,A88,A15;
       reconsider DMN = divset(T.n,i) /\ divset(T.m,j)
         as real-bounded Subset of REAL by XBOOLE_1:17,XXREAL_2:45;
       Snmij - Smnij = (xvol DMN) * (h/.(p1.i) - 0.X - h/.(p2.j))
                    by A83,A84,A89,A85,RLVECT_1:34; then
       Snmij - Smnij = (xvol DMN) * (h/.(p1.i) - (h/.t - h/.t) - h/.(p2.j))
                         by RLVECT_1:15; then
       Snmij - Smnij = (xvol DMN) * (h/.(p1.i) - h/.t + h/.t - h/.(p2.j))
                         by RLVECT_1:29; then
       Snmij - Smnij = (xvol DMN) * ((h/.(p1.i) - h/.t) + (h/.t - h/.(p2.j)))
                         by RLVECT_1:28; then
       Snmij -  Smnij
         = (xvol DMN) * (h/.(p1.i) - h/.t) + (xvol DMN) * (h/.t - h/.(p2.j))
                  by RLVECT_1:def 5; then
A97:   ||. Snmij - Smnij .||
          <= ||. (xvol DMN) * (h/.(p1.i) -h/.t) .||
           + ||. (xvol DMN) * (h/.t - h/.(p2.j)) .|| by NORMSP_1:def 1;
       ||. (xvol DMN) * (h/.(p1.i) - h/.t) .||
          = |. xvol DMN .| * ||. h/.(p1.i) - h/.t .||
     & ||. (xvol DMN) * (h/.t - h/.(p2.j)) .||
          = |. xvol DMN .| * ||. h/.t - h/.(p2.j) .|| by NORMSP_1:def 1; then
A98:  ||. (xvol DMN) * (h/.(p1.i) - h/.t) .||
          = xvol DMN * ||. h/.(p1.i) - h/.t .||
     & ||. (xvol DMN) * (h/.t - h/.(p2.j)) .||
          = xvol DMN * ||. h/.t - h/.(p2.j) .|| by Th5,ABSVALUE:def 1;
       0<= xvol DMN by Th5; then
       ||. (xvol DMN) * (h/.(p1.i) - h/.t) .|| <= (xvol DMN) * p2v
     & ||. (xvol DMN) * (h/.t - h/.(p2.j)) .|| <= (xvol DMN) * p2v
                        by A98,A96,XREAL_1:64; then
       ||. (xvol DMN) * (h/.(p1.i) - h/.t) .||
      + ||. (xvol DMN) * (h/.t - h/.(p2.j)) .||
          <= (xvol DMN) * p2v + (xvol DMN) * p2v by XREAL_1:7;
       hence
        ||. Snmij - Smnij .|| <= (xvol (divset(T.n,i) /\ divset(T.m,j))) *pv
                                   by A97,XXREAL_0:2;
      end;
     end;
A99: for j be Nat st j in dom XZq
       holds ||. XZq/.j .|| <= vol divset(T.m,j) * pv
     proof
      let j be Nat;
      assume A100: j in dom XZq; then
A101:   XZq/.j = Xq/.j - Zq/.j by VFUNCT_1:def 2;
A102:   Xq/.j = Xq.j & Zq/.j = Zq.j by A100,A65,A53,A80,PARTFUN1:def 6;
A103:  dom XZq = dom Xq /\ dom Zq by VFUNCT_1:def 2;
      consider Xsq be FinSequence of X such that
A104:    dom Xsq = Seg len (T.n) & Xq.j = Sum Xsq
     & for i be Nat st i in dom Xsq holds Xsq.i=Snm.(i,j) by A100,A53,A65,A80;
      consider Zsq be FinSequence of X such that
A105:    dom Zsq = Seg len (T.n) & Zq.j = Sum Zsq
     & for i be Nat st i in dom Zsq holds Zsq.i=Smn.(i,j) by A100,A65,A53,A80;
      set XZsq = Xsq - Zsq;
A106:   dom XZsq = dom Xsq /\ dom Zsq by VFUNCT_1:def 2; then
      reconsider XZsq as FinSequence by A104,A105,FINSEQ_1:def 2;
      now let i be Nat;
       assume i in dom XZsq; then
       XZsq.i = (Xsq - Zsq)/.i by PARTFUN1:def 6;
       hence XZsq.i in the carrier of X;
      end; then
      reconsider XZsq as FinSequence of X by FINSEQ_2:12;
      defpred P11[Nat,set] means $2 = xvol (divset(T.n,$1) /\ divset(T.m,j));
A107: for i be Nat st i in Seg len XZsq holds
       ex x be Element of REAL st P11[i,x]
      proof
       let i be Nat;
       assume i in Seg len XZsq;
       xvol (divset(T.n,i) /\ divset(T.m,j)) in REAL by XREAL_0:def 1;
       hence thesis;
      end;
      consider vtntm be FinSequence of REAL such that
A108:  dom vtntm = Seg len XZsq & for i be Nat st i in Seg len XZsq
         holds P11[i,vtntm.i] from FINSEQ_1:sch 5(A107);
A109:  for i be Nat st i in dom vtntm holds
       vtntm.i=xvol ( divset(T.m,j) /\ divset(T.n,i)) by A108;
A110:  len vtntm = len XZsq by A108,FINSEQ_1:def 3;
A111: len XZsq = len (T.n) by A104,A105,A106,FINSEQ_1:def 3;
      reconsider pvtntm = pv*vtntm as FinSequence of REAL;
      j in dom (T.m) by A100,A103,A53,A65,FINSEQ_1:def 3; then
      divset(T.m,j) c= A by INTEGRA1:8; then
      Sum vtntm = vol divset(T.m,j) by Th8,
                           A109,A110,A104,A105,A106,FINSEQ_1:def 3; then
A112: Sum pvtntm = pv * vol divset(T.m,j) by RVSUM_1:87;
      dom pvtntm = dom vtntm by VALUED_1:def 5; then
      dom pvtntm = Seg len (T.n) by A104,A105,A106,A108,FINSEQ_1:def 3; then
A113: len pvtntm =len (T.n) by FINSEQ_1:def 3;
      for i be Nat st i in dom XZsq holds ||. XZsq/.i .|| <= (pvtntm).i
      proof
       let i be Nat;
       assume A114: i in dom XZsq; then
A115:   XZsq/.i = Xsq/.i - Zsq/.i by VFUNCT_1:def 2;
       Xsq/.i = Xsq.i & Zsq/.i = Zsq.i by PARTFUN1:def 6,A114,A105,A106,A104;
 then
A116:   Xsq/.i =Snm.(i,j) & Zsq/.i =Smn.(i,j) by A114,A104,A106,A105;
       dom vtntm = dom XZsq by A108,FINSEQ_1:def 3; then
       pv*(vtntm.i) = pv* (xvol (divset(T.n,i) /\ divset(T.m,j)))
            by A108,A114; then
       (pv(#)vtntm).i = pv* (xvol (divset(T.n,i) /\ divset(T.m,j)))
            by VALUED_1:6;
       hence thesis by A115,A116,A82,A114,A104,A105,A106,A100,A103,A53,A65;
      end; then
A117:  ||. Sum XZsq .|| <= vol (divset(T.m,j)) * pv by A112,Th10,A111,A113;
      len Xsq = len Zsq by FINSEQ_3:29,A104,A105;
      hence thesis by A101,A102,A104,A105,A117,INTEGR18:2;
     end;
     defpred P12[Nat,set] means $2 = vol (divset(T.m,$1));
A118:for i be Nat st i in Seg len XZq holds
       ex x be Element of REAL st P12[i,x]
     proof
      let i be Nat;
      assume i in Seg len XZq;
      vol (divset(T.m,i)) in REAL by XREAL_0:def 1;
      hence thesis;
     end;
     consider vtm  be FinSequence of REAL such that
A119:dom vtm = Seg len XZq & for i be Nat st i in Seg len XZq
        holds P12[i,vtm.i] from FINSEQ_1:sch 5(A118);
A120:len XZq = len (T.m) by A53,A65,A80,FINSEQ_1:def 3;
A121:Seg len XZq = dom XZq by FINSEQ_1:def 3;
A122:Sum vtm = vol A by A119,Th6,A120;
     reconsider pvtm = pv*vtm as FinSequence of REAL;
     dom pvtm = dom vtm by VALUED_1:def 5; then
     dom pvtm = Seg len (T.m) by A53,A65,A80,A119,FINSEQ_1:def 3; then
A123:len pvtm = len (T.m) by FINSEQ_1:def 3;
A124:Sum pvtm = pv * vol A by RVSUM_1:87,A122;
     for j be Nat st j in dom XZq holds ||. XZq/.j .|| <= pvtm.j
     proof
      let j be Nat;
      assume A125: j in dom XZq; then
      vtm.j= vol (divset(T.m,j)) by A119,A121;
      then pvtm.j= vol (divset(T.m,j)) * pv by VALUED_1:6;
      hence ||. XZq/.j .|| <= pvtm.j by A125,A99;
     end;
     then ||. Sum XZq .|| <= (vol A) * pv by A124,A120,A123,Th10;
     hence ||.(middle_sum(h,S).n) - (middle_sum(h,S).m).|| < p
                                by A19,A81,XXREAL_0:2,A14;
    end;
    hence middle_sum(h,S) is convergent by RSSPACE3:8,LOPBAN_1:def 15;
   end;
end;
