integra3.miz



    begin

    reserve a,b,e,r,x,y for Real,

i,j,k,n,m for Element of NAT ,

x1 for set,

p,q for FinSequence of REAL ,

A for non empty closed_interval Subset of REAL ,

D,D1,D2 for Division of A,

f,g for Function of A, REAL ,

T for DivSequence of A;

    

     Lm1: ((j -' j) + 1) = 1

    proof

      (j -' j) = (j - j) by XREAL_1: 233

      .= 0 ;

      hence thesis;

    end;

    

     Lm2: for n st 1 <= n & n <= 2 holds n = 1 or n = 2

    proof

      let n;

      assume that

       A1: 1 <= n and

       A2: n <= 2;

      per cases by A1, XXREAL_0: 1;

        suppose n = 1;

        hence thesis;

      end;

        suppose n > 1;

        then n >= (1 + 1) by NAT_1: 13;

        hence thesis by A2, XXREAL_0: 1;

      end;

    end;

    definition

      let A be non empty closed_interval Subset of REAL , D be Division of A;

      :: INTEGRA3:def1

      func delta (D) -> Real equals ( max ( rng ( upper_volume (( chi (A,A)),D))));

      correctness ;

    end

    definition

      let A be non empty closed_interval Subset of REAL , T be DivSequence of A;

      :: INTEGRA3:def2

      func delta (T) -> Real_Sequence means

      : Def2: for i holds (it . i) = ( delta (T . i));

      existence

      proof

        deffunc F( Nat) = ( delta (T . ( In ($1, NAT ))));

        consider IT be Real_Sequence such that

         A1: for i be Nat holds (IT . i) = F(i) from SEQ_1:sch 1;

        take IT;

        let i;

        (IT . i) = F(i) by A1;

        hence thesis;

      end;

      uniqueness

      proof

        let F1,F2 be Real_Sequence such that

         A2: for i holds (F1 . i) = ( delta (T . i)) and

         A3: for i holds (F2 . i) = ( delta (T . i));

        for i holds (F1 . i) = (F2 . i)

        proof

          let i;

          (F1 . i) = ( delta (T . i)) by A2

          .= (F2 . i) by A3;

          hence thesis;

        end;

        hence thesis by FUNCT_2: 63;

      end;

    end

    theorem :: INTEGRA3:1

    D1 <= D2 implies ( delta D1) >= ( delta D2)

    proof

      ( delta D2) in ( rng ( upper_volume (( chi (A,A)),D2))) by XXREAL_2:def 8;

      then

      consider j such that

       A1: j in ( dom ( upper_volume (( chi (A,A)),D2))) and

       A2: ( delta D2) = (( upper_volume (( chi (A,A)),D2)) . j) by PARTFUN1: 3;

      ( len ( upper_volume (( chi (A,A)),D2))) = ( len D2) by INTEGRA1:def 6;

      then

       A3: j in ( dom D2) by A1, FINSEQ_3: 29;

      then

       A4: ( delta D2) = ( vol ( divset (D2,j))) by A2, INTEGRA1: 20;

      assume D1 <= D2;

      then

      consider i be Nat such that

       A5: i in ( dom D1) and

       A6: ( divset (D2,j)) c= ( divset (D1,i)) by A3, INTEGRA2: 37;

      

       A7: ( vol ( divset (D1,i))) = (( upper_volume (( chi (A,A)),D1)) . i) by A5, INTEGRA1: 20;

      ( len ( upper_volume (( chi (A,A)),D1))) = ( len D1) by INTEGRA1:def 6;

      then i in ( dom ( upper_volume (( chi (A,A)),D1))) by A5, FINSEQ_3: 29;

      then ( vol ( divset (D1,i))) in ( rng ( upper_volume (( chi (A,A)),D1))) by A7, FUNCT_1:def 3;

      then ( delta D2) <= ( max ( rng ( upper_volume (( chi (A,A)),D1)))) by A4, A6, INTEGRA2: 38, XXREAL_2: 61;

      hence thesis;

    end;

    theorem :: INTEGRA3:2

    

     Th2: ( vol A) <> 0 implies ex i st i in ( dom D) & ( vol ( divset (D,i))) > 0

    proof

      assume

       A1: ( vol A) <> 0 ;

      

       A2: ( len D) in ( dom D) by FINSEQ_5: 6;

      assume

       A3: for i st i in ( dom D) holds ( vol ( divset (D,i))) <= 0 ;

      

       A4: i in ( dom D) implies ( vol ( divset (D,i))) = 0

      proof

        assume i in ( dom D);

        then ( vol ( divset (D,i))) <= 0 by A3;

        hence thesis by INTEGRA1: 9;

      end;

      

       A5: i in ( dom D) implies ( upper_bound ( divset (D,i))) = ( lower_bound ( divset (D,i)))

      proof

        assume i in ( dom D);

        then ( vol ( divset (D,i))) = 0 by A4;

        then (( upper_bound ( divset (D,i))) - ( lower_bound ( divset (D,i)))) = 0 by INTEGRA1:def 5;

        hence thesis;

      end;

      

       A6: ( len D) = 1

      proof

        ( len D) < (( len D) + 1) by NAT_1: 13;

        then

         A7: (( len D) - 1) < ( len D) by XREAL_1: 19;

        assume

         A8: ( len D) <> 1;

        then

         A9: ( upper_bound ( divset (D,( len D)))) = (D . ( len D)) by A2, INTEGRA1:def 4;

        

         A10: (( len D) - 1) in ( dom D) by A2, A8, INTEGRA1: 7;

        ( lower_bound ( divset (D,( len D)))) = (D . (( len D) - 1)) by A2, A8, INTEGRA1:def 4;

        then ( lower_bound ( divset (D,( len D)))) < ( upper_bound ( divset (D,( len D)))) by A2, A9, A10, A7, SEQM_3:def 1;

        hence contradiction by A5, A2;

      end;

      then ( upper_bound ( divset (D,( len D)))) = (D . ( len D)) by A2, INTEGRA1:def 4;

      then

       A11: ( upper_bound ( divset (D,( len D)))) = ( upper_bound A) by INTEGRA1:def 2;

      ( lower_bound ( divset (D,( len D)))) = ( lower_bound A) by A2, A6, INTEGRA1:def 4;

      then ( upper_bound A) = (( lower_bound A) + 0 ) by A5, A2, A11;

      then (( upper_bound A) - ( lower_bound A)) = 0 ;

      hence contradiction by A1, INTEGRA1:def 5;

    end;

    theorem :: INTEGRA3:3

    

     Th3: x in A implies ex j st j in ( dom D) & x in ( divset (D,j))

    proof

      assume

       A1: x in A;

      then

       A2: ( lower_bound A) <= x by INTEGRA2: 1;

      

       A3: x <= ( upper_bound A) by A1, INTEGRA2: 1;

      

       A4: ( rng D) <> {} ;

      then

       A5: 1 in ( dom D) by FINSEQ_3: 32;

      per cases ;

        suppose x in ( rng D);

        then

        consider j such that

         A6: j in ( dom D) and

         A7: (D . j) = x by PARTFUN1: 3;

        x in ( divset (D,j))

        proof

          per cases ;

            suppose

             A8: j = 1;

            

             A9: ex a, b st a <= b & a = ( lower_bound ( divset (D,j))) & b = ( upper_bound ( divset (D,j))) by SEQ_4: 11;

            ( upper_bound ( divset (D,j))) = (D . j) by A6, A8, INTEGRA1:def 4;

            hence thesis by A7, A9, INTEGRA2: 1;

          end;

            suppose

             A10: j <> 1;

            

             A11: ex a, b st a <= b & a = ( lower_bound ( divset (D,j))) & b = ( upper_bound ( divset (D,j))) by SEQ_4: 11;

            ( upper_bound ( divset (D,j))) = (D . j) by A6, A10, INTEGRA1:def 4;

            hence thesis by A7, A11, INTEGRA2: 1;

          end;

        end;

        hence thesis by A6;

      end;

        suppose

         A12: not x in ( rng D);

        defpred MIN[ Nat] means x < ( upper_bound ( divset (D,$1))) & $1 in ( dom D);

        

         A13: ( len D) in ( dom D) by FINSEQ_5: 6;

        ( upper_bound ( divset (D,( len D)))) = (D . ( len D))

        proof

          per cases ;

            suppose ( len D) = 1;

            hence thesis by A13, INTEGRA1:def 4;

          end;

            suppose ( len D) <> 1;

            hence thesis by A13, INTEGRA1:def 4;

          end;

        end;

        then

         A14: ( upper_bound ( divset (D,( len D)))) = ( upper_bound A) by INTEGRA1:def 2;

        x <> ( upper_bound A)

        proof

          assume x = ( upper_bound A);

          then x = (D . ( len D)) by INTEGRA1:def 2;

          hence contradiction by A12, A13, FUNCT_1:def 3;

        end;

        then x < ( upper_bound ( divset (D,( len D)))) by A3, A14, XXREAL_0: 1;

        then

         A15: ex k be Nat st MIN[k] by A13;

        consider k be Nat such that

         A16: MIN[k] & for n be Nat st MIN[n] holds k <= n from NAT_1:sch 5( A15);

        defpred MAX[ Nat] means x >= ( lower_bound ( divset (D,$1))) & $1 in ( dom D);

        ( lower_bound ( divset (D,1))) = ( lower_bound A) by A5, INTEGRA1:def 4;

        then

         A17: ex k be Nat st MAX[k] by A2, A4, FINSEQ_3: 32;

        

         A18: for k be Nat holds MAX[k] implies k <= ( len D) by FINSEQ_3: 25;

        consider j be Nat such that

         A19: MAX[j] & for n be Nat st MAX[n] holds n <= j from NAT_1:sch 6( A18, A17);

        k = j

        proof

          assume

           A20: k <> j;

          per cases by A20, XXREAL_0: 1;

            suppose

             A21: k < j;

            

             A22: ( upper_bound ( divset (D,k))) = (D . k)

            proof

              per cases ;

                suppose k = 1;

                hence thesis by A16, INTEGRA1:def 4;

              end;

                suppose k <> 1;

                hence thesis by A16, INTEGRA1:def 4;

              end;

            end;

            

             A23: 1 <= k by A16, FINSEQ_3: 25;

            then (D . (j - 1)) <= x by A19, A21, INTEGRA1:def 4;

            then

             A24: (D . (j - 1)) < (D . k) by A16, A22, XXREAL_0: 2;

            (j - 1) in ( dom D) by A19, A21, A23, INTEGRA1: 7;

            then (j - 1) < k by A16, A24, SEQ_4: 137;

            then j < (k + 1) by XREAL_1: 19;

            hence contradiction by A21, NAT_1: 13;

          end;

            suppose

             A25: k > j;

            x < ( upper_bound ( divset (D,j)))

            proof

              

               A26: ( upper_bound ( divset (D,j))) = (D . j)

              proof

                per cases ;

                  suppose j = 1;

                  hence thesis by A19, INTEGRA1:def 4;

                end;

                  suppose j <> 1;

                  hence thesis by A19, INTEGRA1:def 4;

                end;

              end;

              assume

               A27: x >= ( upper_bound ( divset (D,j)));

              

               A28: (j + 1) <= k by A25, NAT_1: 13;

              

               A29: 1 <= j by A19, FINSEQ_3: 25;

              then

               A30: 1 <= (j + 1) by NAT_1: 13;

              k <= ( len D) by A16, FINSEQ_3: 25;

              then (j + 1) <= ( len D) by A28, XXREAL_0: 2;

              then

               A31: (j + 1) in ( dom D) by A30, FINSEQ_3: 25;

              (j + 1) > 1 by A29, NAT_1: 13;

              

              then ( lower_bound ( divset (D,(j + 1)))) = (D . ((j + 1) - 1)) by A31, INTEGRA1:def 4

              .= (D . j);

              then (j + 1) <= j by A19, A27, A26, A31;

              hence contradiction by NAT_1: 13;

            end;

            hence contradiction by A16, A19, A25;

          end;

        end;

        then x in ( divset (D,k)) by A16, A19, INTEGRA2: 1;

        hence thesis by A16;

      end;

    end;

    theorem :: INTEGRA3:4

    

     Th4: ex D st D1 <= D & D2 <= D & ( rng D) = (( rng D1) \/ ( rng D2))

    proof

      consider D be FinSequence of REAL such that

       A1: ( rng D) = ( rng (D1 ^ D2)) and

       A2: ( len D) = ( card ( rng (D1 ^ D2))) and

       A3: D is increasing by SEQ_4: 140;

      reconsider D as increasing FinSequence of REAL by A3;

      reconsider D as non empty increasing FinSequence of REAL by A1;

      

       A4: ( rng D2) c= A by INTEGRA1:def 2;

      

       A5: ( rng (D1 ^ D2)) = (( rng D1) \/ ( rng D2)) by FINSEQ_1: 31;

      then

       A6: ( rng D1) c= ( rng (D1 ^ D2)) by XBOOLE_1: 7;

      ( rng D1) c= A by INTEGRA1:def 2;

      then

       A7: ( rng D) c= A by A1, A5, A4, XBOOLE_1: 8;

      (D . ( len D)) = ( upper_bound A)

      proof

        ( len D1) in ( dom D1) by FINSEQ_5: 6;

        then (D1 . ( len D1)) in ( rng D1) by FUNCT_1:def 3;

        then

        consider k such that

         A8: k in ( dom D) and

         A9: (D1 . ( len D1)) = (D . k) by A1, A6, PARTFUN1: 3;

        assume

         A10: (D . ( len D)) <> ( upper_bound A);

        

         A11: ( len D) in ( dom D) by FINSEQ_5: 6;

        then (D . ( len D)) in ( rng D) by FUNCT_1:def 3;

        then (D . ( len D)) <= ( upper_bound A) by A7, INTEGRA2: 1;

        then

         A12: (D . ( len D)) < ( upper_bound A) by A10, XXREAL_0: 1;

        (D1 . ( len D1)) = ( upper_bound A) by INTEGRA1:def 2;

        then k > ( len D) by A11, A12, A8, A9, SEQ_4: 137;

        hence contradiction by A8, FINSEQ_3: 25;

      end;

      then

      reconsider D as Division of A by A7, INTEGRA1:def 2;

      take D;

      ( card ( rng D1)) <= ( len D) by A2, A5, NAT_1: 43, XBOOLE_1: 7;

      then ( len D1) <= ( len D) by FINSEQ_4: 62;

      hence D1 <= D by A1, A6, INTEGRA1:def 18;

      

       A13: ( rng D2) c= ( rng (D1 ^ D2)) by A5, XBOOLE_1: 7;

      ( card ( rng D2)) <= ( len D) by A2, A5, NAT_1: 43, XBOOLE_1: 7;

      then ( len D2) <= ( len D) by FINSEQ_4: 62;

      hence D2 <= D by A1, A13, INTEGRA1:def 18;

      thus thesis by A1, FINSEQ_1: 31;

    end;

    theorem :: INTEGRA3:5

    

     Th5: ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D)))) implies for x, y, i st i in ( dom D1) & x in (( rng D) /\ ( divset (D1,i))) & y in (( rng D) /\ ( divset (D1,i))) holds x = y

    proof

      assume

       A1: ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D))));

      let x, y, i;

      assume

       A2: i in ( dom D1);

      assume

       A3: x in (( rng D) /\ ( divset (D1,i)));

      then x in ( rng D) by XBOOLE_0:def 4;

      then

      consider n such that

       A4: n in ( dom D) and

       A5: x = (D . n) by PARTFUN1: 3;

      assume

       A6: y in (( rng D) /\ ( divset (D1,i)));

      then y in ( rng D) by XBOOLE_0:def 4;

      then

      consider m such that

       A7: m in ( dom D) and

       A8: y = (D . m) by PARTFUN1: 3;

      assume

       A9: x <> y;

      

       A10: |.((D . n) - (D . m)).| >= ( min ( rng ( upper_volume (( chi (A,A)),D))))

      proof

        per cases by A9, A5, A8, XXREAL_0: 1;

          suppose n < m;

          then

           A11: (n + 1) <= m by NAT_1: 13;

          

           A12: 1 <= (n + 1) by NAT_1: 12;

          m in ( Seg ( len D)) by A7, FINSEQ_1:def 3;

          then m <= ( len D) by FINSEQ_1: 1;

          then (n + 1) <= ( len D) by A11, XXREAL_0: 2;

          then

           A13: (n + 1) in ( Seg ( len D)) by A12, FINSEQ_1: 1;

          then

           A14: (n + 1) in ( dom D) by FINSEQ_1:def 3;

          then (D . m) >= (D . (n + 1)) by A7, A11, SEQ_4: 137;

          then ((D . n) - (D . m)) <= ((D . n) - (D . (n + 1))) by XREAL_1: 10;

          then

           A15: ( - ((D . n) - (D . m))) >= ( - ((D . n) - (D . (n + 1)))) by XREAL_1: 24;

          (n + 1) in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by A13, INTEGRA1:def 6;

          then (n + 1) in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

          then

           A16: (( upper_volume (( chi (A,A)),D)) . (n + 1)) in ( rng ( upper_volume (( chi (A,A)),D))) by FUNCT_1:def 3;

          n in ( Seg ( len D)) by A4, FINSEQ_1:def 3;

          then 1 <= n by FINSEQ_1: 1;

          then

           A17: (n + 1) <> 1 by NAT_1: 13;

          then

           A18: ( upper_bound ( divset (D,(n + 1)))) = (D . (n + 1)) by A14, INTEGRA1:def 4;

          ( - |.((D . n) - (D . m)).|) <= ((D . n) - (D . m)) by ABSVALUE: 4;

          then

           A19: |.((D . n) - (D . m)).| >= ( - ((D . n) - (D . m))) by XREAL_1: 26;

          ( lower_bound ( divset (D,(n + 1)))) = (D . ((n + 1) - 1)) by A14, A17, INTEGRA1:def 4;

          then ( vol ( divset (D,(n + 1)))) = ((D . (n + 1)) - (D . n)) by A18, INTEGRA1:def 5;

          then ((D . (n + 1)) - (D . n)) = (( upper_volume (( chi (A,A)),D)) . (n + 1)) by A14, INTEGRA1: 20;

          then ((D . (n + 1)) - (D . n)) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A16, XXREAL_2:def 7;

          then ( - ((D . n) - (D . m))) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A15, XXREAL_0: 2;

          hence thesis by A19, XXREAL_0: 2;

        end;

          suppose n > m;

          then

           A20: (m + 1) <= n by NAT_1: 13;

          n in ( Seg ( len D)) by A4, FINSEQ_1:def 3;

          then n <= ( len D) by FINSEQ_1: 1;

          then

           A21: (m + 1) <= ( len D) by A20, XXREAL_0: 2;

          

           A22: 1 <= (m + 1) by NAT_1: 12;

          then

           A23: (m + 1) in ( dom D) by A21, FINSEQ_3: 25;

          then (D . (m + 1)) <= (D . n) by A4, A20, SEQ_4: 137;

          then

           A24: ((D . n) - (D . m)) >= ((D . (m + 1)) - (D . m)) by XREAL_1: 9;

          (m + 1) in ( Seg ( len D)) by A22, A21, FINSEQ_1: 1;

          then (m + 1) in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by INTEGRA1:def 6;

          then (m + 1) in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

          then

           A25: (( upper_volume (( chi (A,A)),D)) . (m + 1)) in ( rng ( upper_volume (( chi (A,A)),D))) by FUNCT_1:def 3;

          m in ( Seg ( len D)) by A7, FINSEQ_1:def 3;

          then 1 <= m by FINSEQ_1: 1;

          then

           A26: 1 < (m + 1) by NAT_1: 13;

          then

           A27: ( upper_bound ( divset (D,(m + 1)))) = (D . (m + 1)) by A23, INTEGRA1:def 4;

          ( lower_bound ( divset (D,(m + 1)))) = (D . ((m + 1) - 1)) by A23, A26, INTEGRA1:def 4;

          then ( vol ( divset (D,(m + 1)))) = ((D . (m + 1)) - (D . m)) by A27, INTEGRA1:def 5;

          then ((D . (m + 1)) - (D . m)) = (( upper_volume (( chi (A,A)),D)) . (m + 1)) by A23, INTEGRA1: 20;

          then ((D . (m + 1)) - (D . m)) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A25, XXREAL_2:def 7;

          then

           A28: ((D . n) - (D . m)) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A24, XXREAL_0: 2;

           |.((D . n) - (D . m)).| >= ((D . n) - (D . m)) by ABSVALUE: 4;

          hence thesis by A28, XXREAL_0: 2;

        end;

      end;

       |.((D . n) - (D . m)).| <= ( delta D1)

      proof

        per cases by A9, A5, A8, XXREAL_0: 1;

          suppose

           A29: n < m;

          i in ( Seg ( len D1)) by A2, FINSEQ_1:def 3;

          then i in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by INTEGRA1:def 6;

          then i in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

          then (( upper_volume (( chi (A,A)),D1)) . i) in ( rng ( upper_volume (( chi (A,A)),D1))) by FUNCT_1:def 3;

          then (( upper_volume (( chi (A,A)),D1)) . i) <= ( max ( rng ( upper_volume (( chi (A,A)),D1)))) by XXREAL_2:def 8;

          then

           A30: (( upper_volume (( chi (A,A)),D1)) . i) <= ( delta D1);

          (D . m) in ( divset (D1,i)) by A6, A8, XBOOLE_0:def 4;

          then (D . m) <= ( upper_bound ( divset (D1,i))) by INTEGRA2: 1;

          then

           A31: ((D . m) - ( lower_bound ( divset (D1,i)))) <= (( upper_bound ( divset (D1,i))) - ( lower_bound ( divset (D1,i)))) by XREAL_1: 9;

          (D . n) in ( divset (D1,i)) by A3, A5, XBOOLE_0:def 4;

          then (D . n) >= ( lower_bound ( divset (D1,i))) by INTEGRA2: 1;

          then ((D . m) - (D . n)) <= ((D . m) - ( lower_bound ( divset (D1,i)))) by XREAL_1: 10;

          then ((D . m) - (D . n)) <= (( upper_bound ( divset (D1,i))) - ( lower_bound ( divset (D1,i)))) by A31, XXREAL_0: 2;

          then ((D . m) - (D . n)) <= ( vol ( divset (D1,i))) by INTEGRA1:def 5;

          then

           A32: ((D . m) - (D . n)) <= (( upper_volume (( chi (A,A)),D1)) . i) by A2, INTEGRA1: 20;

          (D . n) < (D . m) by A4, A7, A29, SEQM_3:def 1;

          then ((D . n) - (D . m)) < 0 by XREAL_1: 49;

          

          then |.((D . n) - (D . m)).| = ( - ((D . n) - (D . m))) by ABSVALUE:def 1

          .= ((D . m) - (D . n));

          hence thesis by A32, A30, XXREAL_0: 2;

        end;

          suppose

           A33: n > m;

          i in ( Seg ( len D1)) by A2, FINSEQ_1:def 3;

          then i in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by INTEGRA1:def 6;

          then i in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

          then (( upper_volume (( chi (A,A)),D1)) . i) in ( rng ( upper_volume (( chi (A,A)),D1))) by FUNCT_1:def 3;

          then (( upper_volume (( chi (A,A)),D1)) . i) <= ( max ( rng ( upper_volume (( chi (A,A)),D1)))) by XXREAL_2:def 8;

          then

           A34: (( upper_volume (( chi (A,A)),D1)) . i) <= ( delta D1);

          (D . n) in ( divset (D1,i)) by A3, A5, XBOOLE_0:def 4;

          then (D . n) <= ( upper_bound ( divset (D1,i))) by INTEGRA2: 1;

          then

           A35: ((D . n) - ( lower_bound ( divset (D1,i)))) <= (( upper_bound ( divset (D1,i))) - ( lower_bound ( divset (D1,i)))) by XREAL_1: 9;

          (D . m) in ( divset (D1,i)) by A6, A8, XBOOLE_0:def 4;

          then (D . m) >= ( lower_bound ( divset (D1,i))) by INTEGRA2: 1;

          then ((D . n) - (D . m)) <= ((D . n) - ( lower_bound ( divset (D1,i)))) by XREAL_1: 10;

          then ((D . n) - (D . m)) <= (( upper_bound ( divset (D1,i))) - ( lower_bound ( divset (D1,i)))) by A35, XXREAL_0: 2;

          then ((D . n) - (D . m)) <= ( vol ( divset (D1,i))) by INTEGRA1:def 5;

          then

           A36: ((D . n) - (D . m)) <= (( upper_volume (( chi (A,A)),D1)) . i) by A2, INTEGRA1: 20;

          (D . n) > (D . m) by A4, A7, A33, SEQM_3:def 1;

          then ((D . n) - (D . m)) > 0 by XREAL_1: 50;

          then |.((D . n) - (D . m)).| = ((D . n) - (D . m)) by ABSVALUE:def 1;

          hence thesis by A36, A34, XXREAL_0: 2;

        end;

      end;

      hence contradiction by A1, A10, XXREAL_0: 2;

    end;

    theorem :: INTEGRA3:6

    

     Th6: for p, q st ( rng p) = ( rng q) & p is increasing & q is increasing holds p = q

    proof

      let p, q;

      assume

       A1: ( rng p) = ( rng q);

      assume that

       A2: p is increasing and

       A3: q is increasing;

      

       A4: q is one-to-one by A3;

      p is one-to-one by A2;

      then ( len p) = ( len q) by A1, A4, FINSEQ_1: 48;

      hence thesis by A1, A2, A3, SEQ_4: 141;

    end;

    theorem :: INTEGRA3:7

    

     Th7: D <= D1 & i in ( dom D) & j in ( dom D) & i <= j implies ( indx (D1,D,i)) <= ( indx (D1,D,j)) & ( indx (D1,D,i)) in ( dom D1)

    proof

      assume that

       A1: D <= D1 and

       A2: i in ( dom D) and

       A3: j in ( dom D) and

       A4: i <= j;

      

       A5: (D . i) = (D1 . ( indx (D1,D,i))) by A1, A2, INTEGRA1:def 19;

      

       A6: ( indx (D1,D,j)) in ( dom D1) by A1, A3, INTEGRA1:def 19;

      

       A7: (D . j) = (D1 . ( indx (D1,D,j))) by A1, A3, INTEGRA1:def 19;

      

       A8: ( indx (D1,D,i)) in ( dom D1) by A1, A2, INTEGRA1:def 19;

      (D . i) <= (D . j) by A2, A3, A4, SEQ_4: 137;

      hence thesis by A5, A8, A7, A6, SEQM_3:def 1;

    end;

    theorem :: INTEGRA3:8

    

     Th8: D <= D1 & i in ( dom D) & j in ( dom D) & i < j implies ( indx (D1,D,i)) < ( indx (D1,D,j))

    proof

      assume that

       A1: D <= D1 and

       A2: i in ( dom D) and

       A3: j in ( dom D) and

       A4: i < j;

      

       A5: (D . i) = (D1 . ( indx (D1,D,i))) by A1, A2, INTEGRA1:def 19;

      

       A6: ( indx (D1,D,j)) in ( dom D1) by A1, A3, INTEGRA1:def 19;

      

       A7: (D . j) = (D1 . ( indx (D1,D,j))) by A1, A3, INTEGRA1:def 19;

      

       A8: ( indx (D1,D,i)) in ( dom D1) by A1, A2, INTEGRA1:def 19;

      (D . i) < (D . j) by A2, A3, A4, SEQM_3:def 1;

      hence thesis by A5, A8, A7, A6, SEQ_4: 137;

    end;

    theorem :: INTEGRA3:9

    

     Th9: ( delta D) >= 0

    proof

      consider y be Element of REAL such that

       A1: y in ( rng D) by SUBSET_1: 4;

      consider n such that

       A2: n in ( dom D) and y = (D . n) by A1, PARTFUN1: 3;

      n in ( Seg ( len D)) by A2, FINSEQ_1:def 3;

      then n in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by INTEGRA1:def 6;

      then n in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

      then (( upper_volume (( chi (A,A)),D)) . n) in ( rng ( upper_volume (( chi (A,A)),D))) by FUNCT_1:def 3;

      then

       A3: (( upper_volume (( chi (A,A)),D)) . n) <= ( max ( rng ( upper_volume (( chi (A,A)),D)))) by XXREAL_2:def 8;

      ( vol ( divset (D,n))) = (( upper_volume (( chi (A,A)),D)) . n) by A2, INTEGRA1: 20;

      then (( upper_volume (( chi (A,A)),D)) . n) >= 0 by INTEGRA1: 9;

      hence thesis by A3;

    end;

    

     Lm3: for A be non empty closed_interval Subset of REAL , g be Function of A, REAL st (g | A) is bounded holds ( upper_bound ( rng g)) >= ( lower_bound ( rng g))

    proof

      let A be non empty closed_interval Subset of REAL ;

      let g be Function of A, REAL ;

      assume

       A1: (g | A) is bounded;

      then

       A2: ( rng g) is bounded_below by INTEGRA1: 11;

      ( rng g) is bounded_above by A1, INTEGRA1: 13;

      hence thesis by A2, SEQ_4: 11;

    end;

    

     Lm4: for A,B be non empty closed_interval Subset of REAL , f be Function of A, REAL st (f | A) is bounded & B c= A holds ( lower_bound ( rng (f | B))) >= ( lower_bound ( rng f)) & ( lower_bound ( rng f)) <= ( upper_bound ( rng (f | B))) & ( upper_bound ( rng (f | B))) <= ( upper_bound ( rng f)) & ( lower_bound ( rng (f | B))) <= ( upper_bound ( rng f))

    proof

      let A,B be non empty closed_interval Subset of REAL , f be Function of A, REAL ;

      assume that

       A1: (f | A) is bounded and

       A2: B c= A;

      B c= ( dom f) by A2, FUNCT_2:def 1;

      then

       A3: ( dom (f | B)) = B by RELAT_1: 62;

      then

       A4: ( rng (f | B)) <> {} by RELAT_1: 42;

      consider x be Element of REAL such that

       A5: x in B by SUBSET_1: 4;

      

       A6: ((f | B) . x) in ( rng (f | B)) by A5, A3, FUNCT_1:def 3;

      

       A7: ( rng f) is bounded_below by A1, INTEGRA1: 11;

      hence

       A8: ( lower_bound ( rng (f | B))) >= ( lower_bound ( rng f)) by A4, RELAT_1: 70, SEQ_4: 47;

      ( rng (f | B)) is bounded_below by A7, RELAT_1: 70, XXREAL_2: 44;

      then

       A9: ( lower_bound ( rng (f | B))) <= ((f | B) . x) by A6, SEQ_4:def 2;

      

       A10: ( rng f) is bounded_above by A1, INTEGRA1: 13;

      then ( rng (f | B)) is bounded_above by RELAT_1: 70, XXREAL_2: 43;

      then ( upper_bound ( rng (f | B))) >= ((f | B) . x) by A6, SEQ_4:def 1;

      then

       A11: ( lower_bound ( rng (f | B))) <= ( upper_bound ( rng (f | B))) by A9, XXREAL_0: 2;

      hence ( upper_bound ( rng (f | B))) >= ( lower_bound ( rng f)) by A8, XXREAL_0: 2;

      thus ( upper_bound ( rng (f | B))) <= ( upper_bound ( rng f)) by A10, A4, RELAT_1: 70, SEQ_4: 48;

      hence thesis by A11, XXREAL_0: 2;

    end;

    

     Lm5: j in ( dom D1) implies ( vol ( divset (D1,j))) <= ( delta D1)

    proof

      assume

       A1: j in ( dom D1);

      then j in ( Seg ( len D1)) by FINSEQ_1:def 3;

      then j in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by INTEGRA1:def 6;

      then j in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

      then (( upper_volume (( chi (A,A)),D1)) . j) in ( rng ( upper_volume (( chi (A,A)),D1))) by FUNCT_1:def 3;

      then (( upper_volume (( chi (A,A)),D1)) . j) <= ( max ( rng ( upper_volume (( chi (A,A)),D1)))) by XXREAL_2:def 8;

      then ( vol ( divset (D1,j))) <= ( max ( rng ( upper_volume (( chi (A,A)),D1)))) by A1, INTEGRA1: 20;

      hence thesis;

    end;

    

     Lm6: for j1 be Element of NAT st j1 = (( len D1) - 1) & x in ( divset (D1,( len D1))) & ( len D1) >= 2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ {x}) holds ( rng (D2 | ( indx (D2,D1,j1)))) = ( rng (D1 | j1))

    proof

      let j1 be Element of NAT ;

      assume that

       A1: j1 = (( len D1) - 1) and

       A2: x in ( divset (D1,( len D1))) and

       A3: ( len D1) >= 2;

      

       A4: ( len D1) in ( dom D1) by FINSEQ_5: 6;

      assume that

       A5: D1 <= D2 and

       A6: ( rng D2) = (( rng D1) \/ {x});

      

       A7: ( len D1) <> 1 by A3;

      then

       A8: (( len D1) - 1) in ( dom D1) by A4, INTEGRA1: 7;

      then

       A9: ( indx (D2,D1,j1)) in ( dom D2) by A1, A5, INTEGRA1:def 19;

      then

       A10: ( indx (D2,D1,j1)) <= ( len D2) by FINSEQ_3: 25;

      

       A11: j1 in ( dom D1) by A1, A4, A7, INTEGRA1: 7;

      then

       A12: 1 <= j1 by FINSEQ_3: 25;

      

       A13: j1 <= ( len D1) by A11, FINSEQ_3: 25;

      ( lower_bound ( divset (D1,( len D1)))) <= x by A2, INTEGRA2: 1;

      then

       A14: (D1 . j1) <= x by A1, A4, A7, INTEGRA1:def 4;

      for x1 be object st x1 in ( rng (D2 | ( indx (D2,D1,j1)))) holds x1 in ( rng (D1 | j1))

      proof

        let x1 be object;

        assume x1 in ( rng (D2 | ( indx (D2,D1,j1))));

        then

        consider k such that

         A15: k in ( dom (D2 | ( indx (D2,D1,j1)))) and

         A16: x1 = ((D2 | ( indx (D2,D1,j1))) . k) by PARTFUN1: 3;

        k in ( Seg ( len (D2 | ( indx (D2,D1,j1))))) by A15, FINSEQ_1:def 3;

        then

         A17: k in ( Seg ( indx (D2,D1,j1))) by A10, FINSEQ_1: 59;

        then

         A18: k in ( dom D2) by A9, RFINSEQ: 6;

        

         A19: ( len (D1 | j1)) = j1 by A13, FINSEQ_1: 59;

        k <= ( indx (D2,D1,j1)) by A17, FINSEQ_1: 1;

        then (D2 . k) <= (D2 . ( indx (D2,D1,j1))) by A9, A18, SEQ_4: 137;

        then

         A20: (D2 . k) <= (D1 . j1) by A1, A5, A8, INTEGRA1:def 19;

        

         A21: (D2 . k) in ( rng D1) implies (D2 . k) in ( rng (D1 | j1))

        proof

          assume (D2 . k) in ( rng D1);

          then

          consider m such that

           A22: m in ( dom D1) and

           A23: (D2 . k) = (D1 . m) by PARTFUN1: 3;

          m in ( Seg ( len D1)) by A22, FINSEQ_1:def 3;

          then

           A24: 1 <= m by FINSEQ_1: 1;

          

           A25: m <= j1 by A11, A20, A22, A23, SEQM_3:def 1;

          then m in ( Seg j1) by A24, FINSEQ_1: 1;

          then

           A26: (D2 . k) = ((D1 | j1) . m) by A11, A23, RFINSEQ: 6;

          m in ( dom (D1 | j1)) by A19, A24, A25, FINSEQ_3: 25;

          hence thesis by A26, FUNCT_1:def 3;

        end;

        

         A27: (D2 . k) in {x} implies (D2 . k) = (D1 . j1)

        proof

          assume (D2 . k) in {x};

          then (D1 . j1) <= (D2 . k) by A14, TARSKI:def 1;

          hence thesis by A20, XXREAL_0: 1;

        end;

        

         A28: (D2 . k) in {x} implies (D2 . k) in ( rng (D1 | j1))

        proof

          j1 in ( dom (D1 | j1)) by A12, A19, FINSEQ_3: 25;

          then

           A29: ((D1 | j1) . j1) in ( rng (D1 | j1)) by FUNCT_1:def 3;

          assume

           A30: (D2 . k) in {x};

          j1 in ( Seg j1) by A12, FINSEQ_1: 1;

          hence thesis by A11, A27, A30, A29, RFINSEQ: 6;

        end;

        (D2 . k) in ( rng D2) by A18, FUNCT_1:def 3;

        hence thesis by A6, A9, A16, A17, A28, A21, RFINSEQ: 6, XBOOLE_0:def 3;

      end;

      then

       A31: ( rng (D2 | ( indx (D2,D1,j1)))) c= ( rng (D1 | j1));

      for x1 be object st x1 in ( rng (D1 | j1)) holds x1 in ( rng (D2 | ( indx (D2,D1,j1))))

      proof

        let x1 be object;

        assume x1 in ( rng (D1 | j1));

        then

        consider k such that

         A32: k in ( dom (D1 | j1)) and

         A33: x1 = ((D1 | j1) . k) by PARTFUN1: 3;

        k in ( Seg ( len (D1 | j1))) by A32, FINSEQ_1:def 3;

        then

         A34: k in ( Seg j1) by A13, FINSEQ_1: 59;

        then

         A35: k in ( dom D1) by A11, RFINSEQ: 6;

        k <= j1 by A34, FINSEQ_1: 1;

        then (D1 . k) <= (D1 . j1) by A1, A8, A35, SEQ_4: 137;

        then (D2 . ( indx (D2,D1,k))) <= (D1 . j1) by A5, A35, INTEGRA1:def 19;

        then

         A36: (D2 . ( indx (D2,D1,k))) <= (D2 . ( indx (D2,D1,j1))) by A1, A5, A8, INTEGRA1:def 19;

        

         A37: ((D1 | j1) . k) = (D1 . k) by A11, A34, RFINSEQ: 6;

        (D1 . k) in ( rng D1) by A35, FUNCT_1:def 3;

        then x1 in ( rng D2) by A6, A33, A37, XBOOLE_0:def 3;

        then

        consider n such that

         A38: n in ( dom D2) and

         A39: x1 = (D2 . n) by PARTFUN1: 3;

        (D2 . ( indx (D2,D1,k))) = (D2 . n) by A5, A33, A37, A35, A39, INTEGRA1:def 19;

        then

         A40: n <= ( indx (D2,D1,j1)) by A9, A38, A36, SEQM_3:def 1;

        1 <= n by A38, FINSEQ_3: 25;

        then

         A41: n in ( Seg ( indx (D2,D1,j1))) by A40, FINSEQ_1: 1;

        then n in ( Seg ( len (D2 | ( indx (D2,D1,j1))))) by A10, FINSEQ_1: 59;

        then

         A42: n in ( dom (D2 | ( indx (D2,D1,j1)))) by FINSEQ_1:def 3;

        (D2 . n) = ((D2 | ( indx (D2,D1,j1))) . n) by A9, A41, RFINSEQ: 6;

        hence thesis by A39, A42, FUNCT_1:def 3;

      end;

      then ( rng (D1 | j1)) c= ( rng (D2 | ( indx (D2,D1,j1))));

      hence thesis by A31, XBOOLE_0:def 10;

    end;

    theorem :: INTEGRA3:10

    

     Th10: x in ( divset (D1,( len D1))) & ( len D1) >= 2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ {x}) & (g | A) is bounded implies (( Sum ( lower_volume (g,D2))) - ( Sum ( lower_volume (g,D1)))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta D1))

    proof

      assume that

       A1: x in ( divset (D1,( len D1))) and

       A2: ( len D1) >= 2;

      set j = ( len D1);

      assume that

       A3: D1 <= D2 and

       A4: ( rng D2) = (( rng D1) \/ {x});

      

       A5: ( len D1) in ( dom D1) by FINSEQ_5: 6;

      then

       A6: ( indx (D2,D1,j)) in ( dom D2) by A3, INTEGRA1:def 19;

      

       A7: ( len D1) <> 1 by A2;

      then

      reconsider j1 = (( len D1) - 1) as Element of NAT by A5, INTEGRA1: 7;

      

       A8: j1 in ( dom D1) by A5, A7, INTEGRA1: 7;

      then

       A9: j1 <= ( len D1) by FINSEQ_3: 25;

      

       A10: 1 <= j1 by A8, FINSEQ_3: 25;

      then ( mid (D1,1,j1)) is increasing by A5, A7, INTEGRA1: 7, INTEGRA1: 35;

      then

       A11: (D1 | j1) is increasing by A10, FINSEQ_6: 116;

      

       A12: (( len D1) - 1) in ( dom D1) by A5, A7, INTEGRA1: 7;

      then

       A13: ( indx (D2,D1,j1)) in ( dom D2) by A3, INTEGRA1:def 19;

      then

       A14: 1 <= ( indx (D2,D1,j1)) by FINSEQ_3: 25;

      then ( mid (D2,1,( indx (D2,D1,j1)))) is increasing by A13, INTEGRA1: 35;

      then

       A15: (D2 | ( indx (D2,D1,j1))) is increasing by A14, FINSEQ_6: 116;

      

       A16: ( indx (D2,D1,j1)) <= ( len D2) by A13, FINSEQ_3: 25;

      then

       A17: ( len (D2 | ( indx (D2,D1,j1)))) = ( indx (D2,D1,j1)) by FINSEQ_1: 59;

      

       A18: ( rng (D2 | ( indx (D2,D1,j1)))) = ( rng (D1 | j1)) by A1, A2, A3, A4, Lm6;

      then

       A19: (D2 | ( indx (D2,D1,j1))) = (D1 | j1) by A15, A11, Th6;

      

       A20: for k st 1 <= k & k <= j1 holds k = ( indx (D2,D1,k))

      proof

        let k;

        assume that

         A21: 1 <= k and

         A22: k <= j1;

        assume

         A23: k <> ( indx (D2,D1,k));

        per cases by A23, XXREAL_0: 1;

          suppose

           A24: k > ( indx (D2,D1,k));

          k <= ( len D1) by A9, A22, XXREAL_0: 2;

          then

           A25: k in ( dom D1) by A21, FINSEQ_3: 25;

          then ( indx (D2,D1,k)) in ( dom D2) by A3, INTEGRA1:def 19;

          then ( indx (D2,D1,k)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

          then

           A26: 1 <= ( indx (D2,D1,k)) by FINSEQ_1: 1;

          

           A27: ( indx (D2,D1,k)) < j1 by A22, A24, XXREAL_0: 2;

          then

           A28: ( indx (D2,D1,k)) in ( Seg j1) by A26, FINSEQ_1: 1;

          ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A3, A8, A22, A25, Th7;

          then ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A26, FINSEQ_1: 1;

          then

           A29: ((D2 | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) = (D2 . ( indx (D2,D1,k))) by A13, RFINSEQ: 6;

          ( indx (D2,D1,k)) <= ( len D1) by A9, A27, XXREAL_0: 2;

          then ( indx (D2,D1,k)) in ( dom D1) by A26, FINSEQ_3: 25;

          then

           A30: (D1 . k) > (D1 . ( indx (D2,D1,k))) by A24, A25, SEQM_3:def 1;

          (D1 . k) = (D2 . ( indx (D2,D1,k))) by A3, A25, INTEGRA1:def 19;

          hence contradiction by A8, A19, A29, A30, A28, RFINSEQ: 6;

        end;

          suppose

           A31: k < ( indx (D2,D1,k));

          k <= ( len D1) by A9, A22, XXREAL_0: 2;

          then

           A32: k in ( dom D1) by A21, FINSEQ_3: 25;

          then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A3, A8, A22, Th7;

          then

           A33: k <= ( indx (D2,D1,j1)) by A31, XXREAL_0: 2;

          then k <= ( len D2) by A16, XXREAL_0: 2;

          then

           A34: k in ( dom D2) by A21, FINSEQ_3: 25;

          k in ( Seg j1) by A21, A22, FINSEQ_1: 1;

          then

           A35: (D1 . k) = ((D1 | j1) . k) by A12, RFINSEQ: 6;

          ( indx (D2,D1,k)) in ( dom D2) by A3, A32, INTEGRA1:def 19;

          then

           A36: (D2 . k) < (D2 . ( indx (D2,D1,k))) by A31, A34, SEQM_3:def 1;

          

           A37: k in ( Seg ( indx (D2,D1,j1))) by A21, A33, FINSEQ_1: 1;

          (D1 . k) = (D2 . ( indx (D2,D1,k))) by A3, A32, INTEGRA1:def 19;

          hence contradiction by A13, A19, A35, A36, A37, RFINSEQ: 6;

        end;

      end;

      

       A38: for k be Nat st 1 <= k & k <= ( len (( lower_volume (g,D1)) | j1)) holds ((( lower_volume (g,D1)) | j1) . k) = ((( lower_volume (g,D2)) | ( indx (D2,D1,j1))) . k)

      proof

        ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A13, FINSEQ_1:def 3;

        then ( indx (D2,D1,j1)) in ( Seg ( len ( lower_volume (g,D2)))) by INTEGRA1:def 7;

        then

         A39: ( indx (D2,D1,j1)) in ( dom ( lower_volume (g,D2))) by FINSEQ_1:def 3;

        let k be Nat;

        assume that

         A40: 1 <= k and

         A41: k <= ( len (( lower_volume (g,D1)) | j1));

        reconsider k as Element of NAT by ORDINAL1:def 12;

        

         A42: ( len ( lower_volume (g,D1))) = ( len D1) by INTEGRA1:def 7;

        then

         A43: k <= j1 by A9, A41, FINSEQ_1: 59;

        then k <= ( len D1) by A9, XXREAL_0: 2;

        then k in ( Seg ( len D1)) by A40, FINSEQ_1: 1;

        then

         A44: k in ( dom D1) by FINSEQ_1:def 3;

        then

         A45: ( indx (D2,D1,k)) in ( dom D2) by A3, INTEGRA1:def 19;

        

         A46: k in ( Seg j1) by A40, A43, FINSEQ_1: 1;

        then ( indx (D2,D1,k)) in ( Seg j1) by A20, A40, A43;

        then

         A47: ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A10, A20;

        then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by FINSEQ_1: 1;

        then

         A48: ( indx (D2,D1,k)) <= ( len D2) by A16, XXREAL_0: 2;

        

         A49: (D1 . k) = (D2 . ( indx (D2,D1,k))) by A3, A44, INTEGRA1:def 19;

        

         A50: ( lower_bound ( divset (D1,k))) = ( lower_bound ( divset (D2,( indx (D2,D1,k))))) & ( upper_bound ( divset (D1,k))) = ( upper_bound ( divset (D2,( indx (D2,D1,k)))))

        proof

          per cases ;

            suppose

             A51: k = 1;

            then

             A52: ( upper_bound ( divset (D1,k))) = (D1 . k) by A44, INTEGRA1:def 4;

            

             A53: ( lower_bound ( divset (D1,k))) = ( lower_bound A) by A44, A51, INTEGRA1:def 4;

            ( indx (D2,D1,k)) = 1 by A10, A20, A51;

            hence thesis by A45, A49, A53, A52, INTEGRA1:def 4;

          end;

            suppose

             A54: k <> 1;

            then

            reconsider k1 = (k - 1) as Element of NAT by A44, INTEGRA1: 7;

            

             A55: (k - 1) in ( dom D1) by A44, A54, INTEGRA1: 7;

            then

             A56: 1 <= k1 by FINSEQ_3: 25;

            k <= (k + 1) by NAT_1: 11;

            then k1 <= k by XREAL_1: 20;

            then

             A57: k1 <= j1 by A43, XXREAL_0: 2;

            

             A58: ( indx (D2,D1,k)) <> 1 by A20, A40, A43, A54;

            then

             A59: ( lower_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . (( indx (D2,D1,k)) - 1)) by A45, INTEGRA1:def 4;

            

             A60: ( upper_bound ( divset (D1,k))) = (D1 . k) by A44, A54, INTEGRA1:def 4;

            

             A61: ( lower_bound ( divset (D1,k))) = (D1 . (k - 1)) by A44, A54, INTEGRA1:def 4;

            

             A62: ( upper_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . ( indx (D2,D1,k))) by A45, A58, INTEGRA1:def 4;

            (D2 . (( indx (D2,D1,k)) - 1)) = (D2 . (k - 1)) by A20, A40, A43

            .= (D2 . ( indx (D2,D1,k1))) by A20, A57, A56;

            hence thesis by A3, A44, A61, A60, A55, A59, A62, INTEGRA1:def 19;

          end;

        end;

        ( divset (D1,k)) = [.( lower_bound ( divset (D1,k))), ( upper_bound ( divset (D1,k))).] by INTEGRA1: 4;

        then

         A63: ( divset (D1,k)) = ( divset (D2,( indx (D2,D1,k)))) by A50, INTEGRA1: 4;

        j1 in ( Seg ( len ( lower_volume (g,D1)))) by A8, A42, FINSEQ_1:def 3;

        then j1 in ( dom ( lower_volume (g,D1))) by FINSEQ_1:def 3;

        

        then

         A64: ((( lower_volume (g,D1)) | j1) . k) = (( lower_volume (g,D1)) . k) by A46, RFINSEQ: 6

        .= (( lower_bound ( rng (g | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A44, A63, INTEGRA1:def 7;

        1 <= ( indx (D2,D1,k)) by A20, A40, A43;

        then

         A65: ( indx (D2,D1,k)) in ( dom D2) by A48, FINSEQ_3: 25;

        ((( lower_volume (g,D2)) | ( indx (D2,D1,j1))) . k) = ((( lower_volume (g,D2)) | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) by A20, A40, A43

        .= (( lower_volume (g,D2)) . ( indx (D2,D1,k))) by A47, A39, RFINSEQ: 6

        .= (( lower_bound ( rng (g | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A65, INTEGRA1:def 7;

        hence thesis by A64;

      end;

      

       A66: ( len D2) in ( dom D2) by FINSEQ_5: 6;

      deffunc LVg( Division of A) = ( lower_volume (g,$1));

      deffunc PLg( Division of A, Nat) = (( PartSums ( lower_volume (g,$1))) . $2);

      

       A67: j >= ( len ( lower_volume (g,D1))) by INTEGRA1:def 7;

      

       A68: j <= ( len LVg(D1)) by INTEGRA1:def 7;

      

       A69: ( len D1) in ( Seg ( len D1)) by FINSEQ_1: 3;

      then

       A70: 1 <= j by FINSEQ_1: 1;

      then

       A71: j in ( dom LVg(D1)) by A68, FINSEQ_3: 25;

      assume

       A72: (g | A) is bounded;

      j < (j + 1) by NAT_1: 13;

      then

       A73: j1 < j by XREAL_1: 19;

      then j1 < ( len LVg(D1)) by INTEGRA1:def 7;

      then j1 in ( dom LVg(D1)) by A10, FINSEQ_3: 25;

      then PLg(D1,j1) = ( Sum ( LVg(D1) | j1)) by INTEGRA1:def 20;

      

      then ( PLg(D1,j1) + ( Sum ( mid ( LVg(D1),j,j)))) = ( Sum (( LVg(D1) | j1) ^ ( mid ( LVg(D1),j,j)))) by RVSUM_1: 75

      .= ( Sum (( mid ( LVg(D1),1,j1)) ^ ( mid ( LVg(D1),(j1 + 1),j)))) by A10, FINSEQ_6: 116

      .= ( Sum ( mid ( LVg(D1),1,j))) by A10, A68, A73, INTEGRA2: 4

      .= ( Sum ( LVg(D1) | j)) by A70, FINSEQ_6: 116;

      then

       A74: ( PLg(D1,j1) + ( Sum ( mid (( lower_volume (g,D1)),j,j)))) = PLg(D1,j) by A71, INTEGRA1:def 20;

      

       A75: ( indx (D2,D1,j)) in ( dom D2) by A3, A5, INTEGRA1:def 19;

      then

       A76: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

      then

       A77: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

      ( len D1) < (( len D1) + 1) by NAT_1: 13;

      then j1 < ( len D1) by XREAL_1: 19;

      then

       A78: ( indx (D2,D1,j1)) < ( indx (D2,D1,( len D1))) by A3, A5, A8, Th8;

      then

       A79: (( indx (D2,D1,j1)) + 1) <= ( indx (D2,D1,( len D1))) by NAT_1: 13;

      

       A80: j1 in ( dom D1) by A5, A7, INTEGRA1: 7;

      

       A81: (( Sum ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,( len D1)))))) - ( Sum ( mid (( lower_volume (g,D1)),( len D1),( len D1))))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta D1))

      proof

        

         A82: (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) <= 2

        proof

          set ID1 = (( indx (D2,D1,j1)) + 1);

          set ID2 = (ID1 + 1);

          assume (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) > 2;

          then

           A83: (( indx (D2,D1,j1)) + (1 + 1)) < ( indx (D2,D1,j)) by XREAL_1: 20;

          

           A84: ID1 < ID2 by NAT_1: 13;

          then ( indx (D2,D1,j1)) <= ID2 by NAT_1: 13;

          then

           A85: 1 <= ID2 by A14, XXREAL_0: 2;

          

           A86: ( indx (D2,D1,j)) in ( dom D2) by A3, A5, INTEGRA1:def 19;

          then

           A87: ( indx (D2,D1,j)) <= ( len D2) by FINSEQ_3: 25;

          then ID2 <= ( len D2) by A83, XXREAL_0: 2;

          then

           A88: ID2 in ( dom D2) by A85, FINSEQ_3: 25;

          then

           A89: (D2 . ID2) < (D2 . ( indx (D2,D1,j))) by A83, A86, SEQM_3:def 1;

          

           A90: 1 <= ID1 by A14, NAT_1: 13;

          

           A91: (D1 . j1) = (D2 . ( indx (D2,D1,j1))) by A3, A8, INTEGRA1:def 19;

          ID1 <= ( indx (D2,D1,j)) by A83, A84, XXREAL_0: 2;

          then ID1 <= ( len D2) by A87, XXREAL_0: 2;

          then

           A92: ID1 in ( dom D2) by A90, FINSEQ_3: 25;

          

           A93: (D1 . j) = (D2 . ( indx (D2,D1,j))) by A3, A5, INTEGRA1:def 19;

          ( indx (D2,D1,j1)) < ID1 by NAT_1: 13;

          then

           A94: (D2 . ( indx (D2,D1,j1))) < (D2 . ID1) by A13, A92, SEQM_3:def 1;

          

           A95: (D2 . ID1) < (D2 . ID2) by A84, A92, A88, SEQM_3:def 1;

          

           A96: not (D2 . ID1) in ( rng D1) & not (D2 . ID2) in ( rng D1)

          proof

            assume

             A97: (D2 . ID1) in ( rng D1) or (D2 . ID2) in ( rng D1);

            per cases by A97;

              suppose (D2 . ID1) in ( rng D1);

              then

              consider n such that

               A98: n in ( dom D1) and

               A99: (D1 . n) = (D2 . ID1) by PARTFUN1: 3;

              j1 < n by A80, A94, A91, A98, A99, SEQ_4: 137;

              then

               A100: j < (n + 1) by XREAL_1: 19;

              (D2 . ID1) < (D2 . ( indx (D2,D1,j))) by A95, A89, XXREAL_0: 2;

              then n < j by A5, A93, A98, A99, SEQ_4: 137;

              hence contradiction by A100, NAT_1: 13;

            end;

              suppose (D2 . ID2) in ( rng D1);

              then

              consider n such that

               A101: n in ( dom D1) and

               A102: (D1 . n) = (D2 . ID2) by PARTFUN1: 3;

              (D2 . ( indx (D2,D1,j1))) < (D2 . ID2) by A94, A95, XXREAL_0: 2;

              then j1 < n by A8, A91, A101, A102, SEQ_4: 137;

              then

               A103: j < (n + 1) by XREAL_1: 19;

              n < j by A5, A89, A93, A101, A102, SEQ_4: 137;

              hence contradiction by A103, NAT_1: 13;

            end;

          end;

          (D2 . ID1) in ( rng D2) by A92, FUNCT_1:def 3;

          then (D2 . ID1) in {x} by A4, A96, XBOOLE_0:def 3;

          then

           A104: (D2 . ID1) = x by TARSKI:def 1;

          (D2 . ID2) in ( rng D2) by A88, FUNCT_1:def 3;

          then (D2 . ID2) in {x} by A4, A96, XBOOLE_0:def 3;

          then (D2 . ID1) = (D2 . ID2) by A104, TARSKI:def 1;

          hence contradiction by A84, A92, A88, SEQ_4: 138;

        end;

        

         A105: j <= ( len ( lower_volume (g,D1))) by INTEGRA1:def 7;

        

         A106: 1 <= j by A69, FINSEQ_1: 1;

        then

         A107: (( mid (( lower_volume (g,D1)),j,j)) . 1) = (( lower_volume (g,D1)) . j) by A105, FINSEQ_6: 118;

        reconsider lv = (( lower_volume (g,D1)) . j) as Element of REAL by XREAL_0:def 1;

        ((j -' j) + 1) = 1 by Lm1;

        then ( len ( mid (( lower_volume (g,D1)),j,j))) = 1 by A106, A105, FINSEQ_6: 118;

        then

         A108: ( mid (( lower_volume (g,D1)),j,j)) = <*lv*> by A107, FINSEQ_1: 40;

        

         A109: 1 <= (( indx (D2,D1,j1)) + 1) by A14, NAT_1: 13;

        ( indx (D2,D1,j)) in ( dom D2) by A3, A5, INTEGRA1:def 19;

        then

         A110: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

        then

         A111: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

        ( indx (D2,D1,j)) in ( Seg ( len ( lower_volume (g,D2)))) by A110, INTEGRA1:def 7;

        then

         A112: ( indx (D2,D1,j)) <= ( len ( lower_volume (g,D2))) by FINSEQ_1: 1;

        then

         A113: (( indx (D2,D1,j1)) + 1) <= ( len ( lower_volume (g,D2))) by A79, XXREAL_0: 2;

        then (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( lower_volume (g,D2)))) by A109, FINSEQ_1: 1;

        then

         A114: (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 7;

        (( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) = (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) by A79, XREAL_1: 233;

        then ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) <= 2 by A82;

        then

         A115: ( len ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) <= 2 by A79, A111, A112, A109, A113, FINSEQ_6: 118;

        ( len ( lower_volume (g,D2))) = ( len D2) by INTEGRA1:def 7;

        then

         A116: (( indx (D2,D1,j1)) + 1) in ( dom D2) by A109, A113, FINSEQ_3: 25;

        ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) >= ( 0 + 1) by XREAL_1: 6;

        then

         A117: 1 <= ( len ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) by A79, A111, A112, A109, A113, FINSEQ_6: 118;

        now

          per cases by A117, A115, Lm2;

            suppose

             A118: ( len ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 1;

            ( upper_bound ( divset (D1,j))) = (D1 . j) by A5, A7, INTEGRA1:def 4;

            then

             A119: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A3, A5, INTEGRA1:def 19;

            ( lower_bound ( divset (D1,j))) = (D1 . j1) by A5, A7, INTEGRA1:def 4;

            then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A3, A8, INTEGRA1:def 19;

            then

             A120: ( divset (D1,j)) = [.(D2 . ( indx (D2,D1,j1))), (D2 . ( indx (D2,D1,j))).] by A119, INTEGRA1: 4;

            

             A121: ( delta D1) >= 0 by Th9;

            

             A122: (( upper_bound ( rng g)) - ( lower_bound ( rng g))) >= 0 by A72, Lm3, XREAL_1: 48;

            

             A123: ( indx (D2,D1,j)) in ( dom D2) by A3, A5, INTEGRA1:def 19;

            ( len ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) by A79, A111, A112, A109, A113, FINSEQ_6: 118;

            then

             A124: (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) = 0 by A79, A118, XREAL_1: 233;

            then ( indx (D2,D1,j)) <> 1 by A13, FINSEQ_3: 25;

            then

             A125: ( upper_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . ( indx (D2,D1,j))) by A123, INTEGRA1:def 4;

            ( lower_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . (( indx (D2,D1,j)) - 1)) by A14, A124, A123, INTEGRA1:def 4;

            then

             A126: ( divset (D2,( indx (D2,D1,j)))) = ( divset (D1,j)) by A124, A120, A125, INTEGRA1: 4;

            reconsider li = (( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) as Element of REAL by XREAL_0:def 1;

            (( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) by A111, A112, A109, A113, FINSEQ_6: 118;

            then ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*li*> by A118, FINSEQ_1: 40;

            

            then ( Sum ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = (( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) by FINSOP_1: 11

            .= (( lower_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))) by A116, INTEGRA1:def 7

            .= (( lower_volume (g,D1)) . j) by A5, A124, A126, INTEGRA1:def 7

            .= ( Sum ( mid (( lower_volume (g,D1)),j,j))) by A108, FINSOP_1: 11;

            hence thesis by A121, A122;

          end;

            suppose

             A127: ( len ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 2;

            

             A128: (( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) by A111, A112, A109, A113, FINSEQ_6: 118;

            

             A129: (2 + (( indx (D2,D1,j1)) + 1)) >= ( 0 + 1) by XREAL_1: 7;

            (( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 2) = ( LVg(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) -' 1)) by A79, A111, A112, A109, A113, A127, FINSEQ_6: 118

            .= ( LVg(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) - 1)) by A129, XREAL_1: 233

            .= ( LVg(D2) . (( indx (D2,D1,j1)) + (1 + 1)));

            then ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*(( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)), (( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 2))*> by A127, A128, FINSEQ_1: 44;

            then

             A130: ( Sum ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) + (( lower_volume (g,D2)) . (( indx (D2,D1,j1)) + 2))) by RVSUM_1: 77;

            

             A131: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) >= 0 by INTEGRA1: 9;

            ( upper_bound ( divset (D1,j))) = (D1 . j) by A5, A7, INTEGRA1:def 4;

            then

             A132: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A3, A5, INTEGRA1:def 19;

            

             A133: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) >= 0 by INTEGRA1: 9;

            ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A79, A111, A112, A109, A113, A127, FINSEQ_6: 118;

            then

             A134: ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A79, XREAL_1: 233;

            then

             A135: (( indx (D2,D1,j1)) + 2) in ( dom D2) by A3, A5, INTEGRA1:def 19;

            ( lower_bound ( divset (D1,j))) = (D1 . j1) by A5, A7, INTEGRA1:def 4;

            then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A3, A8, INTEGRA1:def 19;

            then

             A136: ( vol ( divset (D1,j))) = ((((D2 . (( indx (D2,D1,j1)) + 2)) - (D2 . (( indx (D2,D1,j1)) + 1))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A132, A134, INTEGRA1:def 5;

            (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( lower_volume (g,D2)))) by A109, A113, FINSEQ_1: 1;

            then (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 7;

            then

             A137: (( indx (D2,D1,j1)) + 1) in ( dom D2) by FINSEQ_1:def 3;

            

             A138: (( indx (D2,D1,j1)) + 1) <> 1 by A14, NAT_1: 13;

            then

             A139: ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A137, INTEGRA1:def 4;

            ((( indx (D2,D1,j1)) + 1) - 1) = (( indx (D2,D1,j1)) + 0 );

            then

             A140: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . ( indx (D2,D1,j1))) by A137, A138, INTEGRA1:def 4;

            

             A141: ((( indx (D2,D1,j1)) + 1) + 1) > 1 by A109, NAT_1: 13;

            ((( indx (D2,D1,j1)) + 2) - 1) = (( indx (D2,D1,j1)) + 1);

            then

             A142: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A135, A141, INTEGRA1:def 4;

            ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 2)) by A135, A141, INTEGRA1:def 4;

            

            then ( vol ( divset (D1,j))) = ((( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A142, A136, INTEGRA1:def 5

            .= (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) - ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A140, A139;

            then

             A143: ( vol ( divset (D1,j))) = (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) by INTEGRA1:def 5;

            then

             A144: (( lower_volume (g,D1)) . j) = (( lower_bound ( rng (g | ( divset (D1,j))))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) by A5, INTEGRA1:def 7;

            

             A145: (( Sum ( mid ( LVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) - ( Sum ( mid ( LVg(D1),j,j)))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))))

            proof

              set ID1 = (( indx (D2,D1,j1)) + 1), ID2 = (( indx (D2,D1,j1)) + 2);

              set IR = (( lower_bound ( rng g)) * ( vol ( divset (D2,ID2))));

              ( divset (D1,j)) c= A by A5, INTEGRA1: 8;

              then

               A146: ( lower_bound ( rng (g | ( divset (D1,j))))) >= ( lower_bound ( rng g)) by A72, Lm4;

              ( Sum ( mid ( LVg(D1),j,j))) = ((( lower_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) + (( lower_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A108, A144, FINSOP_1: 11;

              then (( Sum ( mid ( LVg(D1),j,j))) - (( lower_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1))))) >= IR by A133, A146, XREAL_1: 64;

              then ( Sum ( mid ( LVg(D1),j,j))) >= ((( lower_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) + IR) by XREAL_1: 19;

              then

               A147: (( Sum ( mid ( LVg(D1),j,j))) - (( lower_bound ( rng g)) * ( vol ( divset (D2,ID2))))) >= (( lower_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) by XREAL_1: 19;

              (( lower_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) >= (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A131, A146, XREAL_1: 64;

              then (( Sum ( mid ( LVg(D1),j,j))) - (( lower_bound ( rng g)) * ( vol ( divset (D2,ID2))))) >= (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A147, XXREAL_0: 2;

              then

               A148: ( Sum ( mid ( LVg(D1),j,j))) >= (IR + (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1))))) by XREAL_1: 19;

              ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A79, A111, A112, A109, A113, A127, FINSEQ_6: 118;

              then

               A149: ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A79, XREAL_1: 233;

              ID1 in ( dom D2) by A114, FINSEQ_1:def 3;

              then ( divset (D2,ID1)) c= A by INTEGRA1: 8;

              then ( lower_bound ( rng (g | ( divset (D2,ID1))))) <= ( upper_bound ( rng g)) by A72, Lm4;

              then

               A150: (( lower_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1)))) <= (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A131, XREAL_1: 64;

              

               A151: ( indx (D2,D1,j)) in ( dom D2) by A3, A5, INTEGRA1:def 19;

              then ( divset (D2,ID2)) c= A by A149, INTEGRA1: 8;

              then

               A152: ( lower_bound ( rng (g | ( divset (D2,ID2))))) <= ( upper_bound ( rng g)) by A72, Lm4;

              ( Sum ( mid ( LVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( lower_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) + ( LVg(D2) . (( indx (D2,D1,j1)) + 1))) by A130, A151, A149, INTEGRA1:def 7

              .= ((( lower_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) + (( lower_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A116, INTEGRA1:def 7;

              then (( Sum ( mid ( LVg(D2),ID1,( indx (D2,D1,j))))) - (( lower_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1))))) <= (( upper_bound ( rng g)) * ( vol ( divset (D2,ID2)))) by A133, A152, XREAL_1: 64;

              then ( Sum ( mid ( LVg(D2),ID1,( indx (D2,D1,j))))) <= ((( upper_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1))))) by XREAL_1: 20;

              then (( Sum ( mid ( LVg(D2),ID1,( indx (D2,D1,j))))) - (( upper_bound ( rng g)) * ( vol ( divset (D2,ID2))))) <= (( lower_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1)))) by XREAL_1: 20;

              then (( Sum ( mid ( LVg(D2),ID1,( indx (D2,D1,j))))) - (( upper_bound ( rng g)) * ( vol ( divset (D2,ID2))))) <= (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A150, XXREAL_0: 2;

              then ( Sum ( mid ( LVg(D2),ID1,( indx (D2,D1,j))))) <= ((( upper_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1))))) by XREAL_1: 20;

              then (( Sum ( mid ( LVg(D2),ID1,( indx (D2,D1,j))))) - ( Sum ( mid ( LVg(D1),j,j)))) <= (((( upper_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1))))) - (IR + (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1)))))) by A148, XREAL_1: 13;

              hence thesis;

            end;

            (( upper_bound ( rng g)) - ( lower_bound ( rng g))) >= 0 by A72, Lm3, XREAL_1: 48;

            then ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( vol ( divset (D1,j)))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta D1)) by A5, Lm5, XREAL_1: 64;

            hence thesis by A143, A145, XXREAL_0: 2;

          end;

        end;

        hence thesis;

      end;

      ( indx (D2,D1,j1)) in ( dom D2) by A3, A12, INTEGRA1:def 19;

      then ( indx (D2,D1,j1)) <= ( len D2) by FINSEQ_3: 25;

      then

       A153: ( indx (D2,D1,j1)) <= ( len ( lower_volume (g,D2))) by INTEGRA1:def 7;

      j1 <= ( len D1) by A12, FINSEQ_3: 25;

      then

       A154: j1 <= ( len ( lower_volume (g,D1))) by INTEGRA1:def 7;

      

       A155: (D2 . ( indx (D2,D1,j))) = (D1 . ( len D1)) by A3, A5, INTEGRA1:def 19;

      

       A156: ( indx (D2,D1,j)) >= ( len ( lower_volume (g,D2)))

      proof

        assume ( indx (D2,D1,j)) < ( len ( lower_volume (g,D2)));

        then ( indx (D2,D1,j)) < ( len D2) by INTEGRA1:def 7;

        then

         A157: (D1 . ( len D1)) < (D2 . ( len D2)) by A66, A6, A155, SEQM_3:def 1;

        

         A158: not (D2 . ( len D2)) in ( rng D1)

        proof

          assume (D2 . ( len D2)) in ( rng D1);

          (D2 . ( len D2)) <= ( upper_bound A) by INTEGRA1:def 2;

          hence contradiction by A157, INTEGRA1:def 2;

        end;

        (D2 . ( len D2)) in ( rng D2) by A66, FUNCT_1:def 3;

        then (D2 . ( len D2)) in ( rng D1) or (D2 . ( len D2)) in {x} by A4, XBOOLE_0:def 3;

        then (D2 . ( len D2)) = x by A158, TARSKI:def 1;

        then (D2 . ( len D2)) <= ( upper_bound ( divset (D1,( len D1)))) by A1, INTEGRA2: 1;

        hence contradiction by A5, A7, A157, INTEGRA1:def 4;

      end;

      ( indx (D2,D1,j)) in ( Seg ( len D2)) by A6, FINSEQ_1:def 3;

      then ( indx (D2,D1,j)) in ( Seg ( len ( lower_volume (g,D2)))) by INTEGRA1:def 7;

      then ( indx (D2,D1,j)) in ( dom ( lower_volume (g,D2))) by FINSEQ_1:def 3;

      

      then

       A159: PLg(D2,indx) = ( Sum (( lower_volume (g,D2)) | ( indx (D2,D1,j)))) by INTEGRA1:def 20

      .= ( Sum ( lower_volume (g,D2))) by A156, FINSEQ_1: 58;

      j in ( Seg ( len ( lower_volume (g,D1)))) by A69, INTEGRA1:def 7;

      then j in ( dom ( lower_volume (g,D1))) by FINSEQ_1:def 3;

      

      then

       A160: PLg(D1,j) = ( Sum (( lower_volume (g,D1)) | j)) by INTEGRA1:def 20

      .= ( Sum ( lower_volume (g,D1))) by A67, FINSEQ_1: 58;

      ( len D1) = ( len ( lower_volume (g,D1))) by INTEGRA1:def 7;

      then

       A161: j1 in ( dom ( lower_volume (g,D1))) by A8, FINSEQ_3: 29;

      ( len (D2 | ( indx (D2,D1,j1)))) = ( len (D1 | j1)) by A15, A11, A18, Th6;

      then ( indx (D2,D1,j1)) = j1 by A9, A17, FINSEQ_1: 59;

      then ( len (( lower_volume (g,D1)) | j1)) = ( indx (D2,D1,j1)) by A154, FINSEQ_1: 59;

      then ( len (( lower_volume (g,D1)) | j1)) = ( len (( lower_volume (g,D2)) | ( indx (D2,D1,j1)))) by A153, FINSEQ_1: 59;

      then

       A162: (( lower_volume (g,D2)) | ( indx (D2,D1,j1))) = (( lower_volume (g,D1)) | j1) by A38, FINSEQ_1: 14;

      ( len D2) = ( len ( lower_volume (g,D2))) by INTEGRA1:def 7;

      then ( indx (D2,D1,j1)) in ( dom ( lower_volume (g,D2))) by A13, FINSEQ_3: 29;

      

      then

       A163: PLg(D2,indx) = ( Sum (( lower_volume (g,D2)) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20

      .= PLg(D1,j1) by A162, A161, INTEGRA1:def 20;

      ( indx (D2,D1,j)) <= ( len D2) by A76, FINSEQ_1: 1;

      then

       A164: ( indx (D2,D1,j)) <= ( len LVg(D2)) by INTEGRA1:def 7;

      

       A165: ( len D2) = ( len LVg(D2)) by INTEGRA1:def 7;

      then

       A166: ( indx (D2,D1,j)) in ( dom LVg(D2)) by A75, FINSEQ_3: 29;

      ( indx (D2,D1,j1)) in ( dom LVg(D2)) by A13, A165, FINSEQ_3: 29;

      then PLg(D2,indx) = ( Sum ( LVg(D2) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20;

      

      then ( PLg(D2,indx) + ( Sum ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = ( Sum (( LVg(D2) | ( indx (D2,D1,j1))) ^ ( mid ( LVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by RVSUM_1: 75

      .= ( Sum (( mid ( LVg(D2),1,( indx (D2,D1,j1)))) ^ ( mid ( LVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by A14, FINSEQ_6: 116

      .= ( Sum ( mid ( LVg(D2),1,( indx (D2,D1,j))))) by A14, A78, A164, INTEGRA2: 4

      .= ( Sum ( LVg(D2) | ( indx (D2,D1,j)))) by A77, FINSEQ_6: 116;

      then ( PLg(D2,indx) + ( Sum ( mid (( lower_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = PLg(D2,indx) by A166, INTEGRA1:def 20;

      hence thesis by A163, A81, A74, A159, A160;

    end;

    theorem :: INTEGRA3:11

    

     Th11: x in ( divset (D1,( len D1))) & ( len D1) >= 2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ {x}) & (g | A) is bounded implies (( Sum ( upper_volume (g,D1))) - ( Sum ( upper_volume (g,D2)))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta D1))

    proof

      assume that

       A1: x in ( divset (D1,( len D1))) and

       A2: ( len D1) >= 2;

      set j = ( len D1);

      assume that

       A3: D1 <= D2 and

       A4: ( rng D2) = (( rng D1) \/ {x});

      

       A5: ( len D1) in ( Seg ( len D1)) by FINSEQ_1: 3;

      then

       A6: ( len D1) in ( dom D1) by FINSEQ_1:def 3;

      then

       A7: ( indx (D2,D1,j)) in ( dom D2) by A3, INTEGRA1:def 19;

      deffunc UVg( Division of A) = ( upper_volume (g,$1));

      deffunc PUg( Division of A, Nat) = (( PartSums ( upper_volume (g,$1))) . $2);

      

       A8: j >= ( len ( upper_volume (g,D1))) by INTEGRA1:def 6;

      

       A9: ( len D1) <> 1 by A2;

      then

       A10: (( len D1) - 1) in ( dom D1) by A6, INTEGRA1: 7;

      reconsider j1 = (( len D1) - 1) as Element of NAT by A6, A9, INTEGRA1: 7;

      

       A11: ( indx (D2,D1,j1)) in ( dom D2) by A3, A10, INTEGRA1:def 19;

      then

       A12: 1 <= ( indx (D2,D1,j1)) by FINSEQ_3: 25;

      then ( mid (D2,1,( indx (D2,D1,j1)))) is increasing by A11, INTEGRA1: 35;

      then

       A13: (D2 | ( indx (D2,D1,j1))) is increasing by A12, FINSEQ_6: 116;

      ( len D1) < (( len D1) + 1) by NAT_1: 13;

      then j1 < ( len D1) by XREAL_1: 19;

      then

       A14: ( indx (D2,D1,j1)) < ( indx (D2,D1,( len D1))) by A3, A6, A10, Th8;

      then

       A15: (( indx (D2,D1,j1)) + 1) <= ( indx (D2,D1,( len D1))) by NAT_1: 13;

      ( len D2) in ( Seg ( len D2)) by FINSEQ_1: 3;

      then

       A16: ( len D2) in ( dom D2) by FINSEQ_1:def 3;

      

       A17: (D2 . ( indx (D2,D1,j))) = (D1 . ( len D1)) by A3, A6, INTEGRA1:def 19;

      

       A18: ( indx (D2,D1,j)) >= ( len ( upper_volume (g,D2)))

      proof

        assume ( indx (D2,D1,j)) < ( len ( upper_volume (g,D2)));

        then ( indx (D2,D1,j)) < ( len D2) by INTEGRA1:def 6;

        then

         A19: (D1 . ( len D1)) < (D2 . ( len D2)) by A16, A7, A17, SEQM_3:def 1;

        

         A20: not (D2 . ( len D2)) in ( rng D1)

        proof

          assume (D2 . ( len D2)) in ( rng D1);

          (D2 . ( len D2)) <= ( upper_bound A) by INTEGRA1:def 2;

          hence contradiction by A19, INTEGRA1:def 2;

        end;

        (D2 . ( len D2)) in ( rng D2) by A16, FUNCT_1:def 3;

        then (D2 . ( len D2)) in ( rng D1) or (D2 . ( len D2)) in {x} by A4, XBOOLE_0:def 3;

        then (D2 . ( len D2)) = x by A20, TARSKI:def 1;

        then (D2 . ( len D2)) <= ( upper_bound ( divset (D1,( len D1)))) by A1, INTEGRA2: 1;

        hence contradiction by A6, A9, A19, INTEGRA1:def 4;

      end;

      ( indx (D2,D1,j)) in ( Seg ( len D2)) by A7, FINSEQ_1:def 3;

      then ( indx (D2,D1,j)) in ( Seg ( len ( upper_volume (g,D2)))) by INTEGRA1:def 6;

      then ( indx (D2,D1,j)) in ( dom ( upper_volume (g,D2))) by FINSEQ_1:def 3;

      

      then

       A21: PUg(D2,indx) = ( Sum (( upper_volume (g,D2)) | ( indx (D2,D1,j)))) by INTEGRA1:def 20

      .= ( Sum ( upper_volume (g,D2))) by A18, FINSEQ_1: 58;

      ( indx (D2,D1,j)) in ( dom D2) by A3, A6, INTEGRA1:def 19;

      then

       A22: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

      then

       A23: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

      

       A24: ( indx (D2,D1,j1)) <= ( len D2) by A11, FINSEQ_3: 25;

      then

       A25: ( len (D2 | ( indx (D2,D1,j1)))) = ( indx (D2,D1,j1)) by FINSEQ_1: 59;

      

       A26: j1 <= ( len D1) by A10, FINSEQ_3: 25;

      assume

       A27: (g | A) is bounded;

      

       A28: (( Sum ( mid (( upper_volume (g,D1)),( len D1),( len D1)))) - ( Sum ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,( len D1))))))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta D1))

      proof

        

         A29: (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) <= 2

        proof

          reconsider ID1 = (( indx (D2,D1,j1)) + 1) as Element of NAT ;

          reconsider ID2 = (ID1 + 1) as Element of NAT ;

          assume (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) > 2;

          then

           A30: (( indx (D2,D1,j1)) + (1 + 1)) < ( indx (D2,D1,j)) by XREAL_1: 20;

          

           A31: ID1 < ID2 by NAT_1: 13;

          then ( indx (D2,D1,j1)) <= ID2 by NAT_1: 13;

          then

           A32: 1 <= ID2 by A12, XXREAL_0: 2;

          

           A33: ( indx (D2,D1,j)) in ( dom D2) by A3, A6, INTEGRA1:def 19;

          then

           A34: ( indx (D2,D1,j)) <= ( len D2) by FINSEQ_3: 25;

          then ID2 <= ( len D2) by A30, XXREAL_0: 2;

          then

           A35: ID2 in ( dom D2) by A32, FINSEQ_3: 25;

          then

           A36: (D2 . ID2) < (D2 . ( indx (D2,D1,j))) by A30, A33, SEQM_3:def 1;

          

           A37: 1 <= ID1 by A12, NAT_1: 13;

          

           A38: (D1 . j1) = (D2 . ( indx (D2,D1,j1))) by A3, A10, INTEGRA1:def 19;

          ID1 <= ( indx (D2,D1,j)) by A30, A31, XXREAL_0: 2;

          then ID1 <= ( len D2) by A34, XXREAL_0: 2;

          then

           A39: ID1 in ( dom D2) by A37, FINSEQ_3: 25;

          

           A40: (D1 . j) = (D2 . ( indx (D2,D1,j))) by A3, A6, INTEGRA1:def 19;

          ( indx (D2,D1,j1)) < ID1 by NAT_1: 13;

          then

           A41: (D2 . ( indx (D2,D1,j1))) < (D2 . ID1) by A11, A39, SEQM_3:def 1;

          

           A42: (D2 . ID1) < (D2 . ID2) by A31, A39, A35, SEQM_3:def 1;

          

           A43: not (D2 . ID1) in ( rng D1) & not (D2 . ID2) in ( rng D1)

          proof

            assume

             A44: (D2 . ID1) in ( rng D1) or (D2 . ID2) in ( rng D1);

            now

              per cases by A44;

                suppose (D2 . ID1) in ( rng D1);

                then

                consider n such that

                 A45: n in ( dom D1) and

                 A46: (D1 . n) = (D2 . ID1) by PARTFUN1: 3;

                j1 < n by A10, A41, A38, A45, A46, SEQ_4: 137;

                then

                 A47: j < (n + 1) by XREAL_1: 19;

                (D2 . ID1) < (D2 . ( indx (D2,D1,j))) by A42, A36, XXREAL_0: 2;

                then n < j by A6, A40, A45, A46, SEQ_4: 137;

                hence contradiction by A47, NAT_1: 13;

              end;

                suppose (D2 . ID2) in ( rng D1);

                then

                consider n such that

                 A48: n in ( dom D1) and

                 A49: (D1 . n) = (D2 . ID2) by PARTFUN1: 3;

                (D2 . ( indx (D2,D1,j1))) < (D2 . ID2) by A41, A42, XXREAL_0: 2;

                then j1 < n by A10, A38, A48, A49, SEQ_4: 137;

                then

                 A50: j < (n + 1) by XREAL_1: 19;

                n < j by A6, A36, A40, A48, A49, SEQ_4: 137;

                hence contradiction by A50, NAT_1: 13;

              end;

            end;

            hence thesis;

          end;

          (D2 . ID1) in ( rng D2) by A39, FUNCT_1:def 3;

          then (D2 . ID1) in {x} by A4, A43, XBOOLE_0:def 3;

          then

           A51: (D2 . ID1) = x by TARSKI:def 1;

          (D2 . ID2) in ( rng D2) by A35, FUNCT_1:def 3;

          then (D2 . ID2) in {x} by A4, A43, XBOOLE_0:def 3;

          then (D2 . ID1) = (D2 . ID2) by A51, TARSKI:def 1;

          hence contradiction by A31, A39, A35, SEQ_4: 138;

        end;

        

         A52: j <= ( len ( upper_volume (g,D1))) by INTEGRA1:def 6;

        

         A53: 1 <= j by A5, FINSEQ_1: 1;

        then

         A54: (( mid (( upper_volume (g,D1)),j,j)) . 1) = (( upper_volume (g,D1)) . j) by A52, FINSEQ_6: 118;

        reconsider uv = (( upper_volume (g,D1)) . j) as Element of REAL by XREAL_0:def 1;

        ((j -' j) + 1) = 1 by Lm1;

        then ( len ( mid (( upper_volume (g,D1)),j,j))) = 1 by A53, A52, FINSEQ_6: 118;

        then ( mid (( upper_volume (g,D1)),j,j)) = <*uv*> by A54, FINSEQ_1: 40;

        then

         A55: ( Sum ( mid (( upper_volume (g,D1)),j,j))) = (( upper_volume (g,D1)) . j) by FINSOP_1: 11;

        

         A56: 1 <= (( indx (D2,D1,j1)) + 1) by A12, NAT_1: 13;

        ( indx (D2,D1,j)) in ( dom D2) by A3, A6, INTEGRA1:def 19;

        then

         A57: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

        then

         A58: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

        ( indx (D2,D1,j)) in ( Seg ( len ( upper_volume (g,D2)))) by A57, INTEGRA1:def 6;

        then

         A59: ( indx (D2,D1,j)) <= ( len ( upper_volume (g,D2))) by FINSEQ_1: 1;

        then

         A60: (( indx (D2,D1,j1)) + 1) <= ( len ( upper_volume (g,D2))) by A15, XXREAL_0: 2;

        then (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( upper_volume (g,D2)))) by A56, FINSEQ_1: 1;

        then

         A61: (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 6;

        then

         A62: (( indx (D2,D1,j1)) + 1) in ( dom D2) by FINSEQ_1:def 3;

        (( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) = (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) by A15, XREAL_1: 233;

        then ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) <= 2 by A29;

        then

         A63: ( len ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) <= 2 by A15, A58, A59, A56, A60, FINSEQ_6: 118;

        ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) >= ( 0 + 1) by XREAL_1: 6;

        then

         A64: 1 <= ( len ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) by A15, A58, A59, A56, A60, FINSEQ_6: 118;

        now

          per cases by A64, A63, Lm2;

            suppose

             A65: ( len ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 1;

            ( upper_bound ( divset (D1,j))) = (D1 . j) by A6, A9, INTEGRA1:def 4;

            then

             A66: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A3, A6, INTEGRA1:def 19;

            ( lower_bound ( divset (D1,j))) = (D1 . j1) by A6, A9, INTEGRA1:def 4;

            then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A3, A10, INTEGRA1:def 19;

            then

             A67: ( divset (D1,j)) = [.(D2 . ( indx (D2,D1,j1))), (D2 . ( indx (D2,D1,j))).] by A66, INTEGRA1: 4;

            

             A68: ( delta D1) >= 0 by Th9;

            

             A69: (( upper_bound ( rng g)) - ( lower_bound ( rng g))) >= 0 by A27, Lm3, XREAL_1: 48;

            

             A70: ( indx (D2,D1,j)) in ( dom D2) by A3, A6, INTEGRA1:def 19;

            ( len ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) by A15, A58, A59, A56, A60, FINSEQ_6: 118;

            then

             A71: (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) = 0 by A15, A65, XREAL_1: 233;

            then ( indx (D2,D1,j)) <> 1 by A11, FINSEQ_3: 25;

            then

             A72: ( upper_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . ( indx (D2,D1,j))) by A70, INTEGRA1:def 4;

            ( lower_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . (( indx (D2,D1,j)) - 1)) by A12, A71, A70, INTEGRA1:def 4;

            then

             A73: ( divset (D2,( indx (D2,D1,j)))) = ( divset (D1,j)) by A71, A67, A72, INTEGRA1: 4;

            reconsider uv = (( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) as Element of REAL by XREAL_0:def 1;

            (( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) by A58, A59, A56, A60, FINSEQ_6: 118;

            then ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*uv*> by A65, FINSEQ_1: 40;

            

            then ( Sum ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = (( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) by FINSOP_1: 11

            .= (( upper_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))) by A62, INTEGRA1:def 6

            .= ( Sum ( mid (( upper_volume (g,D1)),j,j))) by A6, A55, A71, A73, INTEGRA1:def 6;

            hence thesis by A68, A69;

          end;

            suppose

             A74: ( len ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 2;

            

             A75: (( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) by A58, A59, A56, A60, FINSEQ_6: 118;

            

             A76: (2 + (( indx (D2,D1,j1)) + 1)) >= ( 0 + 1) by XREAL_1: 7;

            (( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 2) = ( UVg(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) -' 1)) by A15, A58, A59, A56, A60, A74, FINSEQ_6: 118

            .= ( UVg(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) - 1)) by A76, XREAL_1: 233

            .= ( UVg(D2) . (( indx (D2,D1,j1)) + (1 + 1)));

            then ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*(( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)), (( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 2))*> by A74, A75, FINSEQ_1: 44;

            then

             A77: ( Sum ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 1)) + (( upper_volume (g,D2)) . (( indx (D2,D1,j1)) + 2))) by RVSUM_1: 77;

            

             A78: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) >= 0 by INTEGRA1: 9;

            ( upper_bound ( divset (D1,j))) = (D1 . j) by A6, A9, INTEGRA1:def 4;

            then

             A79: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A3, A6, INTEGRA1:def 19;

            

             A80: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) >= 0 by INTEGRA1: 9;

            ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A15, A58, A59, A56, A60, A74, FINSEQ_6: 118;

            then

             A81: ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A15, XREAL_1: 233;

            then

             A82: (( indx (D2,D1,j1)) + 2) in ( dom D2) by A3, A6, INTEGRA1:def 19;

            ( lower_bound ( divset (D1,j))) = (D1 . j1) by A6, A9, INTEGRA1:def 4;

            then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A3, A10, INTEGRA1:def 19;

            then

             A83: ( vol ( divset (D1,j))) = ((((D2 . (( indx (D2,D1,j1)) + 2)) - (D2 . (( indx (D2,D1,j1)) + 1))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A79, A81, INTEGRA1:def 5;

            (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( upper_volume (g,D2)))) by A56, A60, FINSEQ_1: 1;

            then (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 6;

            then

             A84: (( indx (D2,D1,j1)) + 1) in ( dom D2) by FINSEQ_1:def 3;

            

             A85: (( indx (D2,D1,j1)) + 1) <> 1 by A12, NAT_1: 13;

            then

             A86: ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A84, INTEGRA1:def 4;

            ((( indx (D2,D1,j1)) + 1) - 1) = (( indx (D2,D1,j1)) + 0 );

            then

             A87: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . ( indx (D2,D1,j1))) by A84, A85, INTEGRA1:def 4;

            

             A88: ((( indx (D2,D1,j1)) + 1) + 1) > 1 by A56, NAT_1: 13;

            ((( indx (D2,D1,j1)) + 2) - 1) = (( indx (D2,D1,j1)) + 1);

            then

             A89: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A82, A88, INTEGRA1:def 4;

            ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 2)) by A82, A88, INTEGRA1:def 4;

            

            then ( vol ( divset (D1,j))) = ((( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A89, A83, INTEGRA1:def 5

            .= (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) - ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A87, A86;

            then

             A90: ( vol ( divset (D1,j))) = (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) by INTEGRA1:def 5;

            then

             A91: (( upper_volume (g,D1)) . j) = (( upper_bound ( rng (g | ( divset (D1,j))))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) by A6, INTEGRA1:def 6;

            

             A92: (( Sum ( mid ( UVg(D1),j,j))) - ( Sum ( mid ( UVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))))

            proof

              set ID1 = (( indx (D2,D1,j1)) + 1), ID2 = (( indx (D2,D1,j1)) + 2);

              

               A93: (( Sum ( mid ( UVg(D1),j,j))) - (( upper_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1))))) = (( upper_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID2)))) by A55, A91;

              ( divset (D1,j)) c= A by A6, INTEGRA1: 8;

              then

               A94: ( upper_bound ( rng (g | ( divset (D1,j))))) <= ( upper_bound ( rng g)) by A27, Lm4;

              then (( upper_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID2)))) <= (( upper_bound ( rng g)) * ( vol ( divset (D2,ID2)))) by A80, XREAL_1: 64;

              then ( Sum ( mid ( UVg(D1),j,j))) <= ((( upper_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) + (( upper_bound ( rng g)) * ( vol ( divset (D2,ID2))))) by A93, XREAL_1: 20;

              then

               A95: (( Sum ( mid ( UVg(D1),j,j))) - (( upper_bound ( rng g)) * ( vol ( divset (D2,ID2))))) <= (( upper_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) by XREAL_1: 20;

              (( upper_bound ( rng (g | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) <= (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A78, A94, XREAL_1: 64;

              then (( Sum ( mid ( UVg(D1),j,j))) - (( upper_bound ( rng g)) * ( vol ( divset (D2,ID2))))) <= (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A95, XXREAL_0: 2;

              then

               A96: ( Sum ( mid ( UVg(D1),j,j))) <= ((( upper_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1))))) by XREAL_1: 20;

              ID1 in ( dom D2) by A61, FINSEQ_1:def 3;

              then ( divset (D2,ID1)) c= A by INTEGRA1: 8;

              then ( upper_bound ( rng (g | ( divset (D2,ID1))))) >= ( lower_bound ( rng g)) by A27, Lm4;

              then

               A97: (( upper_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1)))) >= (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A78, XREAL_1: 64;

              ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A15, A58, A59, A56, A60, A74, FINSEQ_6: 118;

              then

               A98: ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A15, XREAL_1: 233;

              

               A99: ( indx (D2,D1,j)) in ( dom D2) by A3, A6, INTEGRA1:def 19;

              then ( divset (D2,ID2)) c= A by A98, INTEGRA1: 8;

              then

               A100: ( upper_bound ( rng (g | ( divset (D2,ID2))))) >= ( lower_bound ( rng g)) by A27, Lm4;

              ( Sum ( mid ( UVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( upper_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) + ( UVg(D2) . (( indx (D2,D1,j1)) + 1))) by A77, A99, A98, INTEGRA1:def 6

              .= ((( upper_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) + (( upper_bound ( rng (g | ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A62, INTEGRA1:def 6;

              then (( Sum ( mid ( UVg(D2),ID1,( indx (D2,D1,j))))) - (( upper_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1))))) >= (( lower_bound ( rng g)) * ( vol ( divset (D2,ID2)))) by A80, A100, XREAL_1: 64;

              then ( Sum ( mid ( UVg(D2),ID1,( indx (D2,D1,j))))) >= ((( lower_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1))))) by XREAL_1: 19;

              then (( Sum ( mid ( UVg(D2),ID1,( indx (D2,D1,j))))) - (( lower_bound ( rng g)) * ( vol ( divset (D2,ID2))))) >= (( upper_bound ( rng (g | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1)))) by XREAL_1: 19;

              then (( Sum ( mid ( UVg(D2),ID1,( indx (D2,D1,j))))) - (( lower_bound ( rng g)) * ( vol ( divset (D2,ID2))))) >= (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1)))) by A97, XXREAL_0: 2;

              then ( Sum ( mid ( UVg(D2),ID1,( indx (D2,D1,j))))) >= ((( lower_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1))))) by XREAL_1: 19;

              then (( Sum ( mid ( UVg(D1),j,j))) - ( Sum ( mid ( UVg(D2),ID1,( indx (D2,D1,j)))))) <= (((( upper_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng g)) * ( vol ( divset (D2,ID1))))) - ((( lower_bound ( rng g)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng g)) * ( vol ( divset (D2,ID1)))))) by A96, XREAL_1: 13;

              hence thesis;

            end;

            (( upper_bound ( rng g)) - ( lower_bound ( rng g))) >= 0 by A27, Lm3, XREAL_1: 48;

            then ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( vol ( divset (D1,j)))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta D1)) by A6, Lm5, XREAL_1: 64;

            hence thesis by A90, A92, XXREAL_0: 2;

          end;

        end;

        hence thesis;

      end;

      j in ( Seg ( len ( upper_volume (g,D1)))) by A5, INTEGRA1:def 6;

      then j in ( dom ( upper_volume (g,D1))) by FINSEQ_1:def 3;

      

      then

       A101: PUg(D1,j) = ( Sum (( upper_volume (g,D1)) | j)) by INTEGRA1:def 20

      .= ( Sum ( upper_volume (g,D1))) by A8, FINSEQ_1: 58;

      

       A102: j <= ( len UVg(D1)) by INTEGRA1:def 6;

      

       A103: 1 <= j1 by A10, FINSEQ_3: 25;

      then ( mid (D1,1,j1)) is increasing by A6, A9, INTEGRA1: 7, INTEGRA1: 35;

      then

       A104: (D1 | j1) is increasing by A103, FINSEQ_6: 116;

      

       A105: ( rng (D2 | ( indx (D2,D1,j1)))) = ( rng (D1 | j1)) by A1, A2, A3, A4, Lm6;

      then

       A106: (D2 | ( indx (D2,D1,j1))) = (D1 | j1) by A13, A104, Th6;

      

       A107: for k st 1 <= k & k <= j1 holds k = ( indx (D2,D1,k))

      proof

        let k;

        assume that

         A108: 1 <= k and

         A109: k <= j1;

        assume

         A110: k <> ( indx (D2,D1,k));

        now

          per cases by A110, XXREAL_0: 1;

            suppose

             A111: k > ( indx (D2,D1,k));

            k <= ( len D1) by A26, A109, XXREAL_0: 2;

            then

             A112: k in ( dom D1) by A108, FINSEQ_3: 25;

            then ( indx (D2,D1,k)) in ( dom D2) by A3, INTEGRA1:def 19;

            then ( indx (D2,D1,k)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

            then

             A113: 1 <= ( indx (D2,D1,k)) by FINSEQ_1: 1;

            

             A114: ( indx (D2,D1,k)) < j1 by A109, A111, XXREAL_0: 2;

            then

             A115: ( indx (D2,D1,k)) in ( Seg j1) by A113, FINSEQ_1: 1;

            ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A3, A10, A109, A112, Th7;

            then ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A113, FINSEQ_1: 1;

            then

             A116: ((D2 | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) = (D2 . ( indx (D2,D1,k))) by A11, RFINSEQ: 6;

            ( indx (D2,D1,k)) <= ( len D1) by A26, A114, XXREAL_0: 2;

            then ( indx (D2,D1,k)) in ( Seg ( len D1)) by A113, FINSEQ_1: 1;

            then ( indx (D2,D1,k)) in ( dom D1) by FINSEQ_1:def 3;

            then

             A117: (D1 . k) > (D1 . ( indx (D2,D1,k))) by A111, A112, SEQM_3:def 1;

            (D1 . k) = (D2 . ( indx (D2,D1,k))) by A3, A112, INTEGRA1:def 19;

            hence contradiction by A10, A106, A116, A117, A115, RFINSEQ: 6;

          end;

            suppose

             A118: k < ( indx (D2,D1,k));

            k <= ( len D1) by A26, A109, XXREAL_0: 2;

            then

             A119: k in ( dom D1) by A108, FINSEQ_3: 25;

            then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A3, A10, A109, Th7;

            then

             A120: k <= ( indx (D2,D1,j1)) by A118, XXREAL_0: 2;

            then k <= ( len D2) by A24, XXREAL_0: 2;

            then

             A121: k in ( dom D2) by A108, FINSEQ_3: 25;

            k in ( Seg j1) by A108, A109, FINSEQ_1: 1;

            then

             A122: (D1 . k) = ((D1 | j1) . k) by A10, RFINSEQ: 6;

            ( indx (D2,D1,k)) in ( dom D2) by A3, A119, INTEGRA1:def 19;

            then

             A123: (D2 . k) < (D2 . ( indx (D2,D1,k))) by A118, A121, SEQM_3:def 1;

            

             A124: k in ( Seg ( indx (D2,D1,j1))) by A108, A120, FINSEQ_1: 1;

            (D1 . k) = (D2 . ( indx (D2,D1,k))) by A3, A119, INTEGRA1:def 19;

            hence contradiction by A11, A106, A122, A123, A124, RFINSEQ: 6;

          end;

        end;

        hence contradiction;

      end;

      

       A125: for k be Nat st 1 <= k & k <= ( len (( upper_volume (g,D1)) | j1)) holds ((( upper_volume (g,D1)) | j1) . k) = ((( upper_volume (g,D2)) | ( indx (D2,D1,j1))) . k)

      proof

        ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A11, FINSEQ_1:def 3;

        then ( indx (D2,D1,j1)) in ( Seg ( len ( upper_volume (g,D2)))) by INTEGRA1:def 6;

        then

         A126: ( indx (D2,D1,j1)) in ( dom ( upper_volume (g,D2))) by FINSEQ_1:def 3;

        let k be Nat;

        assume that

         A127: 1 <= k and

         A128: k <= ( len (( upper_volume (g,D1)) | j1));

        reconsider k as Element of NAT by ORDINAL1:def 12;

        

         A129: ( len ( upper_volume (g,D1))) = ( len D1) by INTEGRA1:def 6;

        then

         A130: k <= j1 by A26, A128, FINSEQ_1: 59;

        then

         A131: k <= ( len D1) by A26, XXREAL_0: 2;

        then k in ( Seg ( len D1)) by A127, FINSEQ_1: 1;

        then

         A132: k in ( dom D1) by FINSEQ_1:def 3;

        then

         A133: ( indx (D2,D1,k)) in ( dom D2) by A3, INTEGRA1:def 19;

        

         A134: k in ( Seg j1) by A127, A130, FINSEQ_1: 1;

        then ( indx (D2,D1,k)) in ( Seg j1) by A107, A127, A130;

        then

         A135: ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A103, A107;

        then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by FINSEQ_1: 1;

        then

         A136: ( indx (D2,D1,k)) <= ( len D2) by A24, XXREAL_0: 2;

        

         A137: (D1 . k) = (D2 . ( indx (D2,D1,k))) by A3, A132, INTEGRA1:def 19;

        

         A138: ( lower_bound ( divset (D1,k))) = ( lower_bound ( divset (D2,( indx (D2,D1,k))))) & ( upper_bound ( divset (D1,k))) = ( upper_bound ( divset (D2,( indx (D2,D1,k)))))

        proof

          per cases ;

            suppose

             A139: k = 1;

            then

             A140: ( upper_bound ( divset (D1,k))) = (D1 . k) by A132, INTEGRA1:def 4;

            

             A141: ( lower_bound ( divset (D1,k))) = ( lower_bound A) by A132, A139, INTEGRA1:def 4;

            ( indx (D2,D1,k)) = 1 by A103, A107, A139;

            hence thesis by A133, A137, A141, A140, INTEGRA1:def 4;

          end;

            suppose

             A142: k <> 1;

            then

            reconsider k1 = (k - 1) as Element of NAT by A132, INTEGRA1: 7;

            k <= (k + 1) by NAT_1: 11;

            then k1 <= k by XREAL_1: 20;

            then

             A143: k1 <= j1 by A130, XXREAL_0: 2;

            

             A144: (k - 1) in ( dom D1) by A132, A142, INTEGRA1: 7;

            then 1 <= k1 by FINSEQ_3: 25;

            then k1 = ( indx (D2,D1,k1)) by A107, A143;

            then

             A145: (D2 . (( indx (D2,D1,k)) - 1)) = (D2 . ( indx (D2,D1,k1))) by A107, A127, A130;

            

             A146: ( indx (D2,D1,k)) <> 1 by A107, A127, A130, A142;

            then

             A147: ( lower_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . (( indx (D2,D1,k)) - 1)) by A133, INTEGRA1:def 4;

            

             A148: ( upper_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . ( indx (D2,D1,k))) by A133, A146, INTEGRA1:def 4;

            

             A149: ( upper_bound ( divset (D1,k))) = (D1 . k) by A132, A142, INTEGRA1:def 4;

            ( lower_bound ( divset (D1,k))) = (D1 . (k - 1)) by A132, A142, INTEGRA1:def 4;

            hence thesis by A3, A132, A149, A144, A147, A148, A145, INTEGRA1:def 19;

          end;

        end;

        ( divset (D1,k)) = [.( lower_bound ( divset (D1,k))), ( upper_bound ( divset (D1,k))).] by INTEGRA1: 4;

        then

         A150: ( divset (D1,k)) = ( divset (D2,( indx (D2,D1,k)))) by A138, INTEGRA1: 4;

        

         A151: k in ( dom D1) by A127, A131, FINSEQ_3: 25;

        j1 in ( Seg ( len ( upper_volume (g,D1)))) by A10, A129, FINSEQ_1:def 3;

        then j1 in ( dom ( upper_volume (g,D1))) by FINSEQ_1:def 3;

        

        then

         A152: ((( upper_volume (g,D1)) | j1) . k) = (( upper_volume (g,D1)) . k) by A134, RFINSEQ: 6

        .= (( upper_bound ( rng (g | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A151, A150, INTEGRA1:def 6;

        1 <= ( indx (D2,D1,k)) by A107, A127, A130;

        then

         A153: ( indx (D2,D1,k)) in ( dom D2) by A136, FINSEQ_3: 25;

        ((( upper_volume (g,D2)) | ( indx (D2,D1,j1))) . k) = ((( upper_volume (g,D2)) | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) by A107, A127, A130

        .= (( upper_volume (g,D2)) . ( indx (D2,D1,k))) by A135, A126, RFINSEQ: 6

        .= (( upper_bound ( rng (g | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A153, INTEGRA1:def 6;

        hence thesis by A152;

      end;

      ( indx (D2,D1,j1)) in ( dom D2) by A3, A10, INTEGRA1:def 19;

      then ( indx (D2,D1,j1)) <= ( len D2) by FINSEQ_3: 25;

      then

       A154: ( indx (D2,D1,j1)) <= ( len ( upper_volume (g,D2))) by INTEGRA1:def 6;

      j1 in ( Seg ( len D1)) by A10, FINSEQ_1:def 3;

      then j1 <= ( len D1) by FINSEQ_1: 1;

      then

       A155: j1 <= ( len ( upper_volume (g,D1))) by INTEGRA1:def 6;

      ( len (D2 | ( indx (D2,D1,j1)))) = ( len (D1 | j1)) by A13, A104, A105, Th6;

      then ( indx (D2,D1,j1)) = j1 by A26, A25, FINSEQ_1: 59;

      then ( len (( upper_volume (g,D1)) | j1)) = ( indx (D2,D1,j1)) by A155, FINSEQ_1: 59;

      then ( len (( upper_volume (g,D1)) | j1)) = ( len (( upper_volume (g,D2)) | ( indx (D2,D1,j1)))) by A154, FINSEQ_1: 59;

      then

       A156: (( upper_volume (g,D2)) | ( indx (D2,D1,j1))) = (( upper_volume (g,D1)) | j1) by A125, FINSEQ_1: 14;

      j1 in ( Seg ( len D1)) by A10, FINSEQ_1:def 3;

      then j1 in ( Seg ( len ( upper_volume (g,D1)))) by INTEGRA1:def 6;

      then

       A157: j1 in ( dom ( upper_volume (g,D1))) by FINSEQ_1:def 3;

      j < (j + 1) by NAT_1: 13;

      then

       A158: j1 < j by XREAL_1: 19;

      ( indx (D2,D1,j)) <= ( len D2) by A22, FINSEQ_1: 1;

      then

       A159: ( indx (D2,D1,j)) <= ( len UVg(D2)) by INTEGRA1:def 6;

      then

       A160: ( indx (D2,D1,j)) in ( dom UVg(D2)) by A23, FINSEQ_3: 25;

      ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A11, FINSEQ_1:def 3;

      then ( indx (D2,D1,j1)) in ( Seg ( len UVg(D2))) by INTEGRA1:def 6;

      then ( indx (D2,D1,j1)) in ( dom UVg(D2)) by FINSEQ_1:def 3;

      then PUg(D2,indx) = ( Sum ( UVg(D2) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20;

      

      then ( PUg(D2,indx) + ( Sum ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = ( Sum (( UVg(D2) | ( indx (D2,D1,j1))) ^ ( mid ( UVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by RVSUM_1: 75

      .= ( Sum (( mid ( UVg(D2),1,( indx (D2,D1,j1)))) ^ ( mid ( UVg(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by A12, FINSEQ_6: 116

      .= ( Sum ( mid ( UVg(D2),1,( indx (D2,D1,j))))) by A12, A14, A159, INTEGRA2: 4

      .= ( Sum ( UVg(D2) | ( indx (D2,D1,j)))) by A23, FINSEQ_6: 116;

      then

       A161: ( PUg(D2,indx) + ( Sum ( mid (( upper_volume (g,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = PUg(D2,indx) by A160, INTEGRA1:def 20;

      

       A162: 1 <= j by A5, FINSEQ_1: 1;

      then

       A163: j in ( dom UVg(D1)) by A102, FINSEQ_3: 25;

      j1 in ( Seg ( len D1)) by A10, FINSEQ_1:def 3;

      then j1 in ( Seg ( len UVg(D1))) by INTEGRA1:def 6;

      then j1 in ( dom UVg(D1)) by FINSEQ_1:def 3;

      then PUg(D1,j1) = ( Sum ( UVg(D1) | j1)) by INTEGRA1:def 20;

      

      then ( PUg(D1,j1) + ( Sum ( mid ( UVg(D1),j,j)))) = ( Sum (( UVg(D1) | j1) ^ ( mid ( UVg(D1),j,j)))) by RVSUM_1: 75

      .= ( Sum (( mid ( UVg(D1),1,j1)) ^ ( mid ( UVg(D1),(j1 + 1),j)))) by A103, FINSEQ_6: 116

      .= ( Sum ( mid ( UVg(D1),1,j))) by A103, A102, A158, INTEGRA2: 4

      .= ( Sum ( UVg(D1) | j)) by A162, FINSEQ_6: 116;

      then

       A164: ( PUg(D1,j1) + ( Sum ( mid (( upper_volume (g,D1)),j,j)))) = PUg(D1,j) by A163, INTEGRA1:def 20;

      ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A11, FINSEQ_1:def 3;

      then ( indx (D2,D1,j1)) in ( Seg ( len ( upper_volume (g,D2)))) by INTEGRA1:def 6;

      then ( indx (D2,D1,j1)) in ( dom ( upper_volume (g,D2))) by FINSEQ_1:def 3;

      

      then PUg(D2,indx) = ( Sum (( upper_volume (g,D2)) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20

      .= PUg(D1,j1) by A156, A157, INTEGRA1:def 20;

      hence thesis by A28, A161, A164, A21, A101;

    end;

    

     Lm7: ( vol A) <> 0 & y in ( rng ( lower_sum_set f)) implies ex D be Division of A st D in ( dom ( lower_sum_set f)) & y = (( lower_sum_set f) . D) & (D . 1) > ( lower_bound A)

    proof

      assume

       A1: ( vol A) <> 0 ;

      assume y in ( rng ( lower_sum_set f));

      then

      consider D3 be Element of ( divs A) such that

       A2: D3 in ( dom ( lower_sum_set f)) and

       A3: y = (( lower_sum_set f) . D3) by PARTFUN1: 3;

      reconsider D3 as Division of A by INTEGRA1:def 3;

      ( rng D3) <> {} ;

      then

       A4: 1 in ( dom D3) by FINSEQ_3: 32;

      

       A5: ( len D3) in ( Seg ( len D3)) by FINSEQ_1: 3;

      now

        per cases ;

          suppose

           A6: (D3 . 1) <> ( lower_bound A);

          (D3 . 1) in A by A4, INTEGRA1: 6;

          then ( lower_bound A) <= (D3 . 1) by INTEGRA2: 1;

          then (D3 . 1) > ( lower_bound A) by A6, XXREAL_0: 1;

          hence thesis by A2, A3;

        end;

          suppose

           A7: (D3 . 1) = ( lower_bound A);

          ex D be Division of A st D in ( dom ( lower_sum_set f)) & y = (( lower_sum_set f) . D) & (D . 1) > ( lower_bound A)

          proof

            

             A8: (( lower_volume (f,D3)) . 1) = (( lower_bound ( rng (f | ( divset (D3,1))))) * ( vol ( divset (D3,1)))) by A4, INTEGRA1:def 7;

            ( vol A) >= 0 by INTEGRA1: 9;

            then

             A9: (( upper_bound A) - ( lower_bound A)) > 0 by A1, INTEGRA1:def 5;

            

             A10: y = ( lower_sum (f,D3)) by A3, INTEGRA1:def 11

            .= ( Sum ( lower_volume (f,D3))) by INTEGRA1:def 9

            .= ( Sum ((( lower_volume (f,D3)) | 1) ^ (( lower_volume (f,D3)) /^ 1))) by RFINSEQ: 8;

            

             A11: (D3 . ( len D3)) = ( upper_bound A) by INTEGRA1:def 2;

            ( len D3) in ( dom D3) by A5, FINSEQ_1:def 3;

            then

             A12: ( len D3) > 1 by A4, A7, A11, A9, SEQ_4: 137, XREAL_1: 47;

            then

            reconsider D = (D3 /^ 1) as increasing FinSequence of REAL by INTEGRA1: 34;

            

             A13: ( len D) = (( len D3) - 1) by A12, RFINSEQ:def 1;

            ( upper_bound A) > ( lower_bound A) by A9, XREAL_1: 47;

            then ( len D) <> 0 by A7, A13, INTEGRA1:def 2;

            then

            reconsider D as non empty increasing FinSequence of REAL ;

            

             A14: ( len D) in ( dom D) by FINSEQ_5: 6;

            (( len D) + 1) = ( len D3) by A13;

            then

             A15: (D . ( len D)) = ( upper_bound A) by A11, A12, A14, RFINSEQ:def 1;

            

             A16: ( len D3) >= (1 + 1) by A12, NAT_1: 13;

            then

             A17: 2 <= ( len ( lower_volume (f,D3))) by INTEGRA1:def 7;

            (1 + 1) <= ( len D3) by A12, NAT_1: 13;

            then 2 in ( dom D3) by FINSEQ_3: 25;

            then

             A18: (D3 . 1) < (D3 . 2) by A4, SEQM_3:def 1;

            

             A19: ( rng D3) c= A by INTEGRA1:def 2;

            ( rng D) c= ( rng D3) by FINSEQ_5: 33;

            then ( rng D) c= A by A19;

            then

            reconsider D as Division of A by A15, INTEGRA1:def 2;

            

             A20: 1 in ( Seg 1) by FINSEQ_1: 1;

            

             A21: 1 <= ( len ( lower_volume (f,D3))) by A12, INTEGRA1:def 7;

            

             A22: ( len (( lower_volume (f,D3)) | 1)) = 1;

            1 <= ( len ( lower_volume (f,D3))) by A12, INTEGRA1:def 7;

            

            then

             A23: ( len ( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3)))))) = ((( len ( lower_volume (f,D3))) -' 2) + 1) by A17, FINSEQ_6: 118

            .= ((( len D3) -' 2) + 1) by INTEGRA1:def 7

            .= ((( len D3) - 2) + 1) by A16, XREAL_1: 233

            .= (( len D3) - 1);

            

             A24: for i be Nat st 1 <= i & i <= ( len ( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3)))))) holds (( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3))))) . i) = (( lower_volume (f,D)) . i)

            proof

              let i be Nat;

              assume that

               A25: 1 <= i and

               A26: i <= ( len ( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3))))));

              

               A27: 1 <= (i + 1) by NAT_1: 12;

              (i + 1) <= ( len D3) by A23, A26, XREAL_1: 19;

              then

               A28: (i + 1) in ( Seg ( len D3)) by A27, FINSEQ_1: 1;

              then

               A29: (i + 1) in ( dom D3) by FINSEQ_1:def 3;

              

               A30: ( divset (D3,(i + 1))) = ( divset (D,i))

              proof

                

                 A31: (i + 1) in ( dom D3) by A28, FINSEQ_1:def 3;

                

                 A32: 1 <> (i + 1) by A25, NAT_1: 13;

                then

                 A33: ( upper_bound ( divset (D3,(i + 1)))) = (D3 . (i + 1)) by A31, INTEGRA1:def 4;

                

                 A34: i in ( dom D) by A13, A23, A25, A26, FINSEQ_3: 25;

                then

                 A35: (D . i) = (D3 . (i + 1)) by A12, RFINSEQ:def 1;

                

                 A36: ( lower_bound ( divset (D3,(i + 1)))) = (D3 . ((i + 1) - 1)) by A32, A31, INTEGRA1:def 4;

                per cases ;

                  suppose

                   A37: i = 1;

                  then

                   A38: ( upper_bound ( divset (D,i))) = (D . i) by A34, INTEGRA1:def 4;

                  

                   A39: ( lower_bound ( divset (D,i))) = ( lower_bound A) by A34, A37, INTEGRA1:def 4;

                  ( divset (D3,(i + 1))) = [.( lower_bound A), (D . i).] by A7, A33, A36, A35, A37, INTEGRA1: 4;

                  hence thesis by A39, A38, INTEGRA1: 4;

                end;

                  suppose

                   A40: i <> 1;

                  then (i - 1) in ( dom D) by A34, INTEGRA1: 7;

                  

                  then

                   A41: (D . (i - 1)) = (D3 . ((i - 1) + 1)) by A12, RFINSEQ:def 1

                  .= (D3 . i);

                  

                   A42: ( upper_bound ( divset (D,i))) = (D . i) by A34, A40, INTEGRA1:def 4;

                  ( lower_bound ( divset (D,i))) = (D . (i - 1)) by A34, A40, INTEGRA1:def 4;

                  then ( divset (D3,(i + 1))) = [.( lower_bound ( divset (D,i))), ( upper_bound ( divset (D,i))).] by A33, A36, A35, A42, A41, INTEGRA1: 4;

                  hence thesis by INTEGRA1: 4;

                end;

              end;

              i <= (( len ( lower_volume (f,D3))) - 1) by A23, A26, INTEGRA1:def 7;

              then

               A43: i <= ((( len ( lower_volume (f,D3))) - 2) + 1);

              (( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3))))) . i) = (( lower_volume (f,D3)) . ((i + 2) - 1)) by A17, A25, A43, FINSEQ_6: 122

              .= (( lower_volume (f,D3)) . (i + 1));

              then

               A44: (( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3))))) . i) = (( lower_bound ( rng (f | ( divset (D3,(i + 1)))))) * ( vol ( divset (D3,(i + 1))))) by A29, INTEGRA1:def 7;

              i in ( Seg ( len D)) by A13, A23, A25, A26, FINSEQ_1: 1;

              then i in ( dom D) by FINSEQ_1:def 3;

              hence thesis by A44, A30, INTEGRA1:def 7;

            end;

            1 in ( dom ( lower_volume (f,D3))) by A21, FINSEQ_3: 25;

            then ((( lower_volume (f,D3)) | 1) . 1) = (( lower_volume (f,D3)) . 1) by A20, RFINSEQ: 6;

            then

             A45: (( lower_volume (f,D3)) | 1) = <*(( lower_volume (f,D3)) . 1)*> by A22, FINSEQ_1: 40;

            

             A46: (2 -' 1) = (2 - 1) by XREAL_1: 233

            .= 1;

            ( rng D) <> {} ;

            then 1 in ( dom D) by FINSEQ_3: 32;

            

            then

             A47: (D . 1) = (D3 . (1 + 1)) by A12, RFINSEQ:def 1

            .= (D3 . 2);

            D in ( divs A) by INTEGRA1:def 3;

            then

             A48: D in ( dom ( lower_sum_set f)) by FUNCT_2:def 1;

            ( len ( lower_volume (f,D3))) >= 2 by A16, INTEGRA1:def 7;

            then

             A49: ( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3))))) = (( lower_volume (f,D3)) /^ 1) by A46, FINSEQ_6: 117;

            ( len ( mid (( lower_volume (f,D3)),2,( len ( lower_volume (f,D3)))))) = ( len ( lower_volume (f,D))) by A13, A23, INTEGRA1:def 7;

            then

             A50: (( lower_volume (f,D3)) /^ 1) = ( lower_volume (f,D)) by A49, A24, FINSEQ_1: 14;

            ( vol ( divset (D3,1))) = (( upper_bound ( divset (D3,1))) - ( lower_bound ( divset (D3,1)))) by INTEGRA1:def 5

            .= (( upper_bound ( divset (D3,1))) - ( lower_bound A)) by A4, INTEGRA1:def 4

            .= ((D3 . 1) - ( lower_bound A)) by A4, INTEGRA1:def 4

            .= 0 by A7;

            

            then y = ( 0 + ( Sum ( lower_volume (f,D)))) by A10, A45, A8, A50, RVSUM_1: 76

            .= ( lower_sum (f,D)) by INTEGRA1:def 9;

            then y = (( lower_sum_set f) . D) by INTEGRA1:def 11;

            hence thesis by A7, A48, A47, A18;

          end;

          hence thesis;

        end;

      end;

      hence thesis;

    end;

    theorem :: INTEGRA3:12

    

     Th12: i in ( dom D) & j in ( dom D) & i <= j & r < (( mid (D,i,j)) . 1) implies ex B be non empty closed_interval Subset of REAL st r = ( lower_bound B) & ( upper_bound B) = (( mid (D,i,j)) . ( len ( mid (D,i,j)))) & ( mid (D,i,j)) is Division of B

    proof

      assume

       A1: i in ( dom D);

      assume

       A2: j in ( dom D);

      assume i <= j;

      then

      consider C be non empty closed_interval Subset of REAL such that

       A3: ( lower_bound C) = (( mid (D,i,j)) . 1) and

       A4: ( upper_bound C) = (( mid (D,i,j)) . ( len ( mid (D,i,j)))) and

       A5: ( mid (D,i,j)) is Division of C by A1, A2, INTEGRA1: 36;

      reconsider MD = ( mid (D,i,j)) as non empty increasing FinSequence of REAL by A5;

      assume

       A6: r < (( mid (D,i,j)) . 1);

      reconsider rr = r, ub = ( upper_bound C) as Real;

      ex a, b st a <= b & a = ( lower_bound C) & b = ( upper_bound C) by SEQ_4: 11;

      then r <= ( upper_bound C) by A6, A3, XXREAL_0: 2;

      then

      reconsider B = [.rr, ub.] as non empty closed_interval Subset of REAL by MEASURE5: 14;

      

       A7: B = [.( lower_bound B), ( upper_bound B).] by INTEGRA1: 4;

      then

       A8: ( lower_bound B) = r by INTEGRA1: 5;

      

       A9: ( upper_bound B) = ( upper_bound C) by A7, INTEGRA1: 5;

      for x be Element of REAL holds x in C implies x in B

      proof

        let x be Element of REAL ;

        assume

         A10: x in C;

        then ( lower_bound C) <= x by INTEGRA2: 1;

        then

         A11: r <= x by A6, A3, XXREAL_0: 2;

        x <= ( upper_bound C) by A10, INTEGRA2: 1;

        hence thesis by A8, A9, A11, INTEGRA2: 1;

      end;

      then

       A12: C c= B;

      ( rng ( mid (D,i,j))) c= C by A5, INTEGRA1:def 2;

      then ( rng ( mid (D,i,j))) c= B by A12;

      then MD is Division of B by A4, A9, INTEGRA1:def 2;

      hence thesis by A4, A8, A9;

    end;

    

     Lm8: ( vol A) <> 0 & ( len D1) = 1 implies ( <*( lower_bound A)*> ^ D1) is non empty increasing FinSequence of REAL

    proof

      assume

       A1: ( vol A) <> 0 ;

      reconsider lb = ( lower_bound A) as Element of REAL by XREAL_0:def 1;

      set MD1 = ( <*lb*> ^ D1);

      

       A2: ( vol A) >= 0 by INTEGRA1: 9;

      assume ( len D1) = 1;

      then (D1 . 1) = ( upper_bound A) by INTEGRA1:def 2;

      then

       A3: ((D1 . 1) - ( lower_bound A)) > 0 by A1, A2, INTEGRA1:def 5;

      then

       A4: ( lower_bound A) < (D1 . 1) by XREAL_1: 47;

      for n,m be Nat holds n in ( dom MD1) & m in ( dom MD1) & n < m implies (MD1 . n) < (MD1 . m)

      proof

        let n,m be Nat;

        assume that

         A5: n in ( dom MD1) and

         A6: m in ( dom MD1) and

         A7: n < m;

        

         A8: not m in ( dom <*( lower_bound A)*>)

        proof

          assume m in ( dom <*( lower_bound A)*>);

          then m in ( Seg ( len <*( lower_bound A)*>)) by FINSEQ_1:def 3;

          then m in {1} by FINSEQ_1: 2, FINSEQ_1: 39;

          then

           A9: n < 1 by A7, TARSKI:def 1;

          n in ( Seg ( len MD1)) by A5, FINSEQ_1:def 3;

          hence contradiction by A9, FINSEQ_1: 1;

        end;

        

         A10: not (MD1 . m) in ( rng <*( lower_bound A)*>)

        proof

          assume (MD1 . m) in ( rng <*( lower_bound A)*>);

          then (MD1 . m) in {( lower_bound A)} by FINSEQ_1: 38;

          then

           A11: (MD1 . m) = ( lower_bound A) by TARSKI:def 1;

          ( rng D1) <> {} ;

          then

           A12: 1 in ( dom D1) by FINSEQ_3: 32;

          consider n be Nat such that

           A13: n in ( dom D1) and

           A14: m = (( len <*( lower_bound A)*>) + n) by A6, A8, FINSEQ_1: 25;

          n in ( Seg ( len D1)) by A13, FINSEQ_1:def 3;

          then

           A15: 1 <= n by FINSEQ_1: 1;

          (D1 . n) = (MD1 . m) by A13, A14, FINSEQ_1:def 7;

          hence contradiction by A3, A11, A13, A15, A12, SEQ_4: 137, XREAL_1: 47;

        end;

        (MD1 . m) in ( rng MD1) by A6, FUNCT_1:def 3;

        then (MD1 . m) in (( rng <*( lower_bound A)*>) \/ ( rng D1)) by FINSEQ_1: 31;

        then

         A16: (MD1 . m) in ( rng D1) by A10, XBOOLE_0:def 3;

        now

          per cases by A5, FINSEQ_1: 25;

            suppose

             A17: n in ( dom <*( lower_bound A)*>);

            then n in ( Seg ( len <*( lower_bound A)*>)) by FINSEQ_1:def 3;

            then n in {1} by FINSEQ_1: 2, FINSEQ_1: 39;

            then

             A18: n = 1 by TARSKI:def 1;

            

             A19: (MD1 . n) = ( <*( lower_bound A)*> . n) by A17, FINSEQ_1:def 7

            .= ( lower_bound A) by A18, FINSEQ_1:def 8;

            ( rng D1) <> {} ;

            then

             A20: 1 in ( dom D1) by FINSEQ_3: 32;

            consider k such that

             A21: k in ( dom D1) and

             A22: (MD1 . m) = (D1 . k) by A16, PARTFUN1: 3;

            1 <= k by A21, FINSEQ_3: 25;

            then (D1 . 1) <= (MD1 . m) by A21, A22, A20, SEQ_4: 137;

            hence thesis by A4, A19, XXREAL_0: 2;

          end;

            suppose ex i be Nat st i in ( dom D1) & n = (( len <*( lower_bound A)*>) + i);

            then

            consider i such that

             A23: i in ( dom D1) and

             A24: n = (( len <*( lower_bound A)*>) + i);

            

             A25: (D1 . i) = (MD1 . n) by A23, A24, FINSEQ_1:def 7;

            consider j be Nat such that

             A26: j in ( dom D1) and

             A27: m = (( len <*( lower_bound A)*>) + j) by A6, A8, FINSEQ_1: 25;

            

             A28: (D1 . j) = (MD1 . m) by A26, A27, FINSEQ_1:def 7;

            i < j by A7, A24, A27, XREAL_1: 6;

            hence thesis by A23, A26, A25, A28, SEQM_3:def 1;

          end;

        end;

        hence thesis;

      end;

      hence thesis by SEQM_3:def 1;

    end;

    

     Lm9: ( lower_bound A) < (D2 . 1) implies ( <*( lower_bound A)*> ^ D2) is non empty increasing FinSequence of REAL

    proof

      reconsider lb = ( lower_bound A) as Element of REAL by XREAL_0:def 1;

      set MD2 = ( <*lb*> ^ D2);

      assume

       A1: ( lower_bound A) < (D2 . 1);

      for n,m be Nat holds n in ( dom MD2) & m in ( dom MD2) & n < m implies (MD2 . n) < (MD2 . m)

      proof

        let n,m be Nat;

        assume that

         A2: n in ( dom MD2) and

         A3: m in ( dom MD2) and

         A4: n < m;

        

         A5: not m in ( dom <*( lower_bound A)*>)

        proof

          assume m in ( dom <*( lower_bound A)*>);

          then m in ( Seg ( len <*( lower_bound A)*>)) by FINSEQ_1:def 3;

          then m in {1} by FINSEQ_1: 2, FINSEQ_1: 39;

          then

           A6: n < 1 by A4, TARSKI:def 1;

          n in ( Seg ( len MD2)) by A2, FINSEQ_1:def 3;

          hence contradiction by A6, FINSEQ_1: 1;

        end;

        

         A7: not (MD2 . m) in ( rng <*( lower_bound A)*>)

        proof

          assume (MD2 . m) in ( rng <*( lower_bound A)*>);

          then (MD2 . m) in {( lower_bound A)} by FINSEQ_1: 38;

          then

           A8: (MD2 . m) = ( lower_bound A) by TARSKI:def 1;

          ( rng D2) <> {} ;

          then

           A9: 1 in ( dom D2) by FINSEQ_3: 32;

          consider n be Nat such that

           A10: n in ( dom D2) and

           A11: m = (( len <*( lower_bound A)*>) + n) by A3, A5, FINSEQ_1: 25;

          n in ( Seg ( len D2)) by A10, FINSEQ_1:def 3;

          then

           A12: 1 <= n by FINSEQ_1: 1;

          (D2 . n) = (MD2 . m) by A10, A11, FINSEQ_1:def 7;

          hence contradiction by A1, A8, A10, A12, A9, SEQ_4: 137;

        end;

        (MD2 . m) in ( rng MD2) by A3, FUNCT_1:def 3;

        then (MD2 . m) in (( rng <*( lower_bound A)*>) \/ ( rng D2)) by FINSEQ_1: 31;

        then

         A13: (MD2 . m) in ( rng <*( lower_bound A)*>) or (MD2 . m) in ( rng D2) by XBOOLE_0:def 3;

        now

          per cases by A2, FINSEQ_1: 25;

            suppose

             A14: n in ( dom <*( lower_bound A)*>);

            then n in ( Seg ( len <*( lower_bound A)*>)) by FINSEQ_1:def 3;

            then n in {1} by FINSEQ_1: 2, FINSEQ_1: 39;

            then

             A15: n = 1 by TARSKI:def 1;

            

             A16: (MD2 . n) = ( <*( lower_bound A)*> . n) by A14, FINSEQ_1:def 7

            .= ( lower_bound A) by A15, FINSEQ_1:def 8;

            ( rng D2) <> {} ;

            then

             A17: 1 in ( dom D2) by FINSEQ_3: 32;

            consider k such that

             A18: k in ( dom D2) and

             A19: (MD2 . m) = (D2 . k) by A13, A7, PARTFUN1: 3;

            k in ( Seg ( len D2)) by A18, FINSEQ_1:def 3;

            then 1 <= k by FINSEQ_1: 1;

            then (D2 . 1) <= (MD2 . m) by A18, A19, A17, SEQ_4: 137;

            hence thesis by A1, A16, XXREAL_0: 2;

          end;

            suppose ex i be Nat st i in ( dom D2) & n = (( len <*( lower_bound A)*>) + i);

            then

            consider i such that

             A20: i in ( dom D2) and

             A21: n = (( len <*( lower_bound A)*>) + i);

            

             A22: (D2 . i) = (MD2 . n) by A20, A21, FINSEQ_1:def 7;

            consider j be Nat such that

             A23: j in ( dom D2) and

             A24: m = (( len <*( lower_bound A)*>) + j) by A3, A5, FINSEQ_1: 25;

            

             A25: (D2 . j) = (MD2 . m) by A23, A24, FINSEQ_1:def 7;

            i < j by A4, A21, A24, XREAL_1: 6;

            hence thesis by A20, A23, A22, A25, SEQM_3:def 1;

          end;

        end;

        hence thesis;

      end;

      hence thesis by SEQM_3:def 1;

    end;

    theorem :: INTEGRA3:13

    

     Lm10: for MD1 be Division of A holds MD1 = ( <*( lower_bound A)*> ^ D1) implies (for i st i in ( Seg ( len D1)) holds ( divset (MD1,(i + 1))) = ( divset (D1,i))) & ( upper_volume (f,D1)) = (( upper_volume (f,MD1)) /^ 1) & ( lower_volume (f,D1)) = (( lower_volume (f,MD1)) /^ 1)

    proof

      let MD1 be Division of A;

      assume

       A1: MD1 = ( <*( lower_bound A)*> ^ D1);

      thus

       A2: for i st i in ( Seg ( len D1)) holds ( divset (MD1,(i + 1))) = ( divset (D1,i))

      proof

        let i;

        assume

         A3: i in ( Seg ( len D1));

        then

         A4: i in ( dom D1) by FINSEQ_1:def 3;

        i <= ( len D1) by A3, FINSEQ_1: 1;

        then (i + 1) <= (( len D1) + 1) by XREAL_1: 6;

        then (i + 1) <= (( len D1) + ( len <*( lower_bound A)*>)) by FINSEQ_1: 39;

        then

         A5: (i + 1) <= ( len MD1) by A1, FINSEQ_1: 22;

        1 <= (i + 1) by NAT_1: 11;

        then

         A6: (i + 1) in ( dom MD1) by A5, FINSEQ_3: 25;

        

         A7: 1 <= i by A3, FINSEQ_1: 1;

        

         A8: ( lower_bound ( divset (D1,i))) = ( lower_bound ( divset (MD1,(i + 1)))) & ( upper_bound ( divset (D1,i))) = ( upper_bound ( divset (MD1,(i + 1))))

        proof

          per cases ;

            suppose

             A9: i = 1;

            

             A10: (i + 1) > 1 by A7, NAT_1: 13;

            then ( lower_bound ( divset (MD1,(i + 1)))) = (MD1 . ((i + 1) - 1)) by A6, INTEGRA1:def 4;

            then

             A11: ( lower_bound ( divset (MD1,(i + 1)))) = ( lower_bound A) by A1, A9, FINSEQ_1: 41;

            

             A12: (MD1 . (i + 1)) = (MD1 . (i + ( len <*( lower_bound A)*>))) by FINSEQ_1: 40

            .= (D1 . i) by A1, A4, FINSEQ_1:def 7;

            ( upper_bound ( divset (MD1,(i + 1)))) = (MD1 . (i + 1)) by A6, A10, INTEGRA1:def 4;

            hence thesis by A4, A9, A11, A12, INTEGRA1:def 4;

          end;

            suppose

             A13: i <> 1;

            

             A14: (i + 1) > 1 by A7, NAT_1: 13;

            (MD1 . (i + 1)) = (MD1 . (i + ( len <*( lower_bound A)*>))) by FINSEQ_1: 40

            .= (D1 . i) by A1, A4, FINSEQ_1:def 7;

            

            then

             A15: ( upper_bound ( divset (MD1,(i + 1)))) = (D1 . i) by A6, A14, INTEGRA1:def 4

            .= ( upper_bound ( divset (D1,i))) by A4, A13, INTEGRA1:def 4;

            (i - 1) in ( dom D1) by A4, A13, INTEGRA1: 7;

            

            then (D1 . (i - 1)) = (MD1 . ((i - 1) + ( len <*( lower_bound A)*>))) by A1, FINSEQ_1:def 7

            .= (MD1 . ((i - 1) + 1)) by FINSEQ_1: 39

            .= (MD1 . ((i + 1) - 1));

            

            then ( lower_bound ( divset (D1,i))) = (MD1 . ((i + 1) - 1)) by A4, A13, INTEGRA1:def 4

            .= ( lower_bound ( divset (MD1,(i + 1)))) by A6, A14, INTEGRA1:def 4;

            hence thesis by A15;

          end;

        end;

        ( divset (D1,i)) = [.( lower_bound ( divset (D1,i))), ( upper_bound ( divset (D1,i))).] by INTEGRA1: 4;

        hence thesis by A8, INTEGRA1: 4;

      end;

      

       A16: ( len MD1) = (( len <*( lower_bound A)*>) + ( len D1)) by A1, FINSEQ_1: 22

      .= (1 + ( len D1)) by FINSEQ_1: 39;

      thus ( upper_volume (f,D1)) = (( upper_volume (f,MD1)) /^ 1)

      proof

        set D2 = D1, MD2 = MD1;

        ( rng ( upper_volume (f,MD2))) <> {} ;

        then 1 in ( dom ( upper_volume (f,MD2))) by FINSEQ_3: 32;

        then 1 <= ( len ( upper_volume (f,MD2))) by FINSEQ_3: 25;

        

        then ( len (( upper_volume (f,MD2)) /^ 1)) = (( len ( upper_volume (f,MD2))) - 1) by RFINSEQ:def 1

        .= (( len MD2) - 1) by INTEGRA1:def 6

        .= ( len D2) by A16;

        then

         A17: ( len ( upper_volume (f,D2))) = ( len (( upper_volume (f,MD2)) /^ 1)) by INTEGRA1:def 6;

        for k be Nat holds 1 <= k & k <= ( len ( upper_volume (f,D2))) implies (( upper_volume (f,D2)) . k) = ((( upper_volume (f,MD2)) /^ 1) . k)

        proof

          let k be Nat;

          assume that

           A18: 1 <= k and

           A19: k <= ( len ( upper_volume (f,D2)));

          (k + 1) <= (( len ( upper_volume (f,D2))) + 1) by A19, XREAL_1: 6;

          then

           A20: (k + 1) <= (( len D2) + 1) by INTEGRA1:def 6;

          k in ( Seg ( len ( upper_volume (f,D2)))) by A18, A19, FINSEQ_1: 1;

          then

           A21: k in ( Seg ( len D2)) by INTEGRA1:def 6;

          then k in ( dom D2) by FINSEQ_1:def 3;

          

          then

           A22: (( upper_volume (f,D2)) . k) = (( upper_bound ( rng (f | ( divset (D2,k))))) * ( vol ( divset (D2,k)))) by INTEGRA1:def 6

          .= (( upper_bound ( rng (f | ( divset (MD2,(k + 1)))))) * ( vol ( divset (D2,k)))) by A2, A21

          .= (( upper_bound ( rng (f | ( divset (MD2,(k + 1)))))) * ( vol ( divset (MD2,(k + 1))))) by A2, A21;

          

           A23: ( len (( upper_volume (f,MD2)) /^ 1)) <= ( len ( upper_volume (f,MD2))) by FINSEQ_5: 25;

          1 <= (k + 1) by NAT_1: 11;

          then (k + 1) in ( Seg ( len MD2)) by A16, A20, FINSEQ_1: 1;

          then

           A24: (k + 1) in ( dom MD2) by FINSEQ_1:def 3;

          1 <= ( len ( upper_volume (f,D2))) by A18, A19, XXREAL_0: 2;

          then

           A25: 1 <= ( len ( upper_volume (f,MD2))) by A17, A23, XXREAL_0: 2;

          k in ( dom (( upper_volume (f,MD2)) /^ 1)) by A17, A18, A19, FINSEQ_3: 25;

          

          then ((( upper_volume (f,MD2)) /^ 1) . k) = (( upper_volume (f,MD2)) . (k + 1)) by A25, RFINSEQ:def 1

          .= (( upper_bound ( rng (f | ( divset (MD2,(k + 1)))))) * ( vol ( divset (MD2,(k + 1))))) by A24, INTEGRA1:def 6;

          hence thesis by A22;

        end;

        hence thesis by A17, FINSEQ_1: 14;

      end;

      ( rng ( lower_volume (f,MD1))) <> {} ;

      then 1 in ( dom ( lower_volume (f,MD1))) by FINSEQ_3: 32;

      then 1 <= ( len ( lower_volume (f,MD1))) by FINSEQ_3: 25;

      

      then ( len (( lower_volume (f,MD1)) /^ 1)) = (( len ( lower_volume (f,MD1))) - 1) by RFINSEQ:def 1

      .= (( len MD1) - 1) by INTEGRA1:def 7

      .= ( len D1) by A16;

      then

       A26: ( len ( lower_volume (f,D1))) = ( len (( lower_volume (f,MD1)) /^ 1)) by INTEGRA1:def 7;

      for k be Nat holds 1 <= k & k <= ( len ( lower_volume (f,D1))) implies (( lower_volume (f,D1)) . k) = ((( lower_volume (f,MD1)) /^ 1) . k)

      proof

        let k be Nat;

        assume that

         A27: 1 <= k and

         A28: k <= ( len ( lower_volume (f,D1)));

        

         A29: 1 <= (k + 1) by NAT_1: 11;

        k in ( Seg ( len ( lower_volume (f,D1)))) by A27, A28, FINSEQ_1: 1;

        then

         A30: k in ( Seg ( len D1)) by INTEGRA1:def 7;

        then k in ( dom D1) by FINSEQ_1:def 3;

        

        then

         A31: (( lower_volume (f,D1)) . k) = (( lower_bound ( rng (f | ( divset (D1,k))))) * ( vol ( divset (D1,k)))) by INTEGRA1:def 7

        .= (( lower_bound ( rng (f | ( divset (MD1,(k + 1)))))) * ( vol ( divset (D1,k)))) by A2, A30

        .= (( lower_bound ( rng (f | ( divset (MD1,(k + 1)))))) * ( vol ( divset (MD1,(k + 1))))) by A2, A30;

        

         A32: ( len (( lower_volume (f,MD1)) /^ 1)) <= ( len ( lower_volume (f,MD1))) by FINSEQ_5: 25;

        (k + 1) <= (( len ( lower_volume (f,D1))) + 1) by A28, XREAL_1: 6;

        then

         A33: (k + 1) <= (( len D1) + 1) by INTEGRA1:def 7;

        ( len MD1) = (( len <*( lower_bound A)*>) + ( len D1)) by A1, FINSEQ_1: 22

        .= (( len D1) + 1) by FINSEQ_1: 39;

        then (k + 1) in ( Seg ( len MD1)) by A29, A33, FINSEQ_1: 1;

        then

         A34: (k + 1) in ( dom MD1) by FINSEQ_1:def 3;

        1 <= ( len (( lower_volume (f,MD1)) /^ 1)) by A26, A27, A28, XXREAL_0: 2;

        then

         A35: 1 <= ( len ( lower_volume (f,MD1))) by A32, XXREAL_0: 2;

        k in ( dom (( lower_volume (f,MD1)) /^ 1)) by A26, A27, A28, FINSEQ_3: 25;

        

        then ((( lower_volume (f,MD1)) /^ 1) . k) = (( lower_volume (f,MD1)) . (k + 1)) by A35, RFINSEQ:def 1

        .= (( lower_bound ( rng (f | ( divset (MD1,(k + 1)))))) * ( vol ( divset (MD1,(k + 1))))) by A34, INTEGRA1:def 7;

        hence thesis by A31;

      end;

      hence thesis by A26, FINSEQ_1: 14;

    end;

    

     Lm11: for MD2 be Division of A st MD2 = ( <*( lower_bound A)*> ^ D2) holds ( vol ( divset (MD2,1))) = 0

    proof

      let MD2 be Division of A;

      assume

       A1: MD2 = ( <*( lower_bound A)*> ^ D2);

      ( rng MD2) <> {} ;

      then

       A2: 1 in ( dom MD2) by FINSEQ_3: 32;

      then

       A3: ( upper_bound ( divset (MD2,1))) = (MD2 . 1) by INTEGRA1:def 4;

      ( lower_bound ( divset (MD2,1))) = ( lower_bound A) by A2, INTEGRA1:def 4;

      

      then ( vol ( divset (MD2,1))) = ((MD2 . 1) - ( lower_bound A)) by A3, INTEGRA1:def 5

      .= (( lower_bound A) - ( lower_bound A)) by A1, FINSEQ_1: 41;

      hence thesis;

    end;

    

     Lm12: for MD1 be Division of A holds MD1 = ( <*( lower_bound A)*> ^ D1) implies ( delta MD1) = ( delta D1)

    proof

      let MD1 be Division of A;

      assume

       A1: MD1 = ( <*( lower_bound A)*> ^ D1);

      then

       A2: ( vol ( divset (MD1,1))) = 0 by Lm11;

      ( delta D1) in ( rng ( upper_volume (( chi (A,A)),D1))) by XXREAL_2:def 8;

      then

      consider i such that

       A3: i in ( dom ( upper_volume (( chi (A,A)),D1))) and

       A4: (( upper_volume (( chi (A,A)),D1)) . i) = ( delta D1) by PARTFUN1: 3;

      ( delta MD1) in ( rng ( upper_volume (( chi (A,A)),MD1))) by XXREAL_2:def 8;

      then

      consider j such that

       A5: j in ( dom ( upper_volume (( chi (A,A)),MD1))) and

       A6: (( upper_volume (( chi (A,A)),MD1)) . j) = ( delta MD1) by PARTFUN1: 3;

      j in ( Seg ( len ( upper_volume (( chi (A,A)),MD1)))) by A5, FINSEQ_1:def 3;

      then

       A7: j in ( Seg ( len MD1)) by INTEGRA1:def 6;

      then

       A8: j in ( dom MD1) by FINSEQ_1:def 3;

      then

       A9: ( delta MD1) = (( upper_bound ( rng (( chi (A,A)) | ( divset (MD1,j))))) * ( vol ( divset (MD1,j)))) by A6, INTEGRA1:def 6;

      

       A10: ( delta MD1) <= ( delta D1)

      proof

        per cases ;

          suppose j = 1;

          hence thesis by A2, A9, Th9;

        end;

          suppose j <> 1;

          then not j in ( Seg 1) by FINSEQ_1: 2, TARSKI:def 1;

          then not j in ( Seg ( len <*( lower_bound A)*>)) by FINSEQ_1: 39;

          then

           A11: not j in ( dom <*( lower_bound A)*>) by FINSEQ_1:def 3;

          j in ( dom MD1) by A7, FINSEQ_1:def 3;

          then

          consider k be Nat such that

           A12: k in ( dom D1) and

           A13: j = (( len <*( lower_bound A)*>) + k) by A1, A11, FINSEQ_1: 25;

          

           A14: k in ( Seg ( len D1)) by A12, FINSEQ_1:def 3;

          

          then ( divset (D1,k)) = ( divset (MD1,(k + 1))) by A1, Lm10

          .= ( divset (MD1,j)) by A13, FINSEQ_1: 39;

          then ( delta MD1) = (( upper_bound ( rng (( chi (A,A)) | ( divset (D1,k))))) * ( vol ( divset (D1,k)))) by A6, A8, INTEGRA1:def 6;

          then

           A15: ( delta MD1) = (( upper_volume (( chi (A,A)),D1)) . k) by A12, INTEGRA1:def 6;

          k in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by A14, INTEGRA1:def 6;

          then k in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

          then ( delta MD1) in ( rng ( upper_volume (( chi (A,A)),D1))) by A15, FUNCT_1:def 3;

          hence thesis by XXREAL_2:def 8;

        end;

      end;

      i in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by A3, FINSEQ_1:def 3;

      then

       A16: i in ( Seg ( len D1)) by INTEGRA1:def 6;

      then i in ( dom D1) by FINSEQ_1:def 3;

      then (( len <*( lower_bound A)*>) + i) in ( dom MD1) by A1, FINSEQ_1: 28;

      then

       A17: (i + 1) in ( dom MD1) by FINSEQ_1: 39;

      then (i + 1) in ( Seg ( len MD1)) by FINSEQ_1:def 3;

      then (i + 1) in ( Seg ( len ( upper_volume (( chi (A,A)),MD1)))) by INTEGRA1:def 6;

      then

       A18: (i + 1) in ( dom ( upper_volume (( chi (A,A)),MD1))) by FINSEQ_1:def 3;

      i in ( dom D1) by A16, FINSEQ_1:def 3;

      

      then ( delta D1) = (( upper_bound ( rng (( chi (A,A)) | ( divset (D1,i))))) * ( vol ( divset (D1,i)))) by A4, INTEGRA1:def 6

      .= (( upper_bound ( rng (( chi (A,A)) | ( divset (MD1,(i + 1)))))) * ( vol ( divset (D1,i)))) by A1, A16, Lm10

      .= (( upper_bound ( rng (( chi (A,A)) | ( divset (MD1,(i + 1)))))) * ( vol ( divset (MD1,(i + 1))))) by A1, A16, Lm10;

      then ( delta D1) = (( upper_volume (( chi (A,A)),MD1)) . (i + 1)) by A17, INTEGRA1:def 6;

      then ( delta D1) in ( rng ( upper_volume (( chi (A,A)),MD1))) by A18, FUNCT_1:def 3;

      then ( delta D1) <= ( delta MD1) by XXREAL_2:def 8;

      hence thesis by A10, XXREAL_0: 1;

    end;

    theorem :: INTEGRA3:14

    

     Th13: x in ( divset (D1,( len D1))) & ( vol A) <> 0 & D1 <= D2 & ( rng D2) = (( rng D1) \/ {x}) & (f | A) is bounded & x > ( lower_bound A) implies (( Sum ( lower_volume (f,D2))) - ( Sum ( lower_volume (f,D1)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))

    proof

      assume that

       A1: x in ( divset (D1,( len D1))) and

       A2: ( vol A) <> 0 and

       A3: D1 <= D2 and

       A4: ( rng D2) = (( rng D1) \/ {x}) and

       A5: (f | A) is bounded and

       A6: x > ( lower_bound A);

      ( len D1) in ( Seg ( len D1)) by FINSEQ_1: 3;

      then

       A7: 1 <= ( len D1) by FINSEQ_1: 1;

      then ( len D1) = 1 or ( len D1) > 1 by XXREAL_0: 1;

      then

       A8: ( len D1) = 1 or ( len D1) >= (1 + 1) by NAT_1: 13;

      now

        per cases by A8;

          suppose

           A9: ( len D1) = 1;

          then

          reconsider MD1 = ( <*( lower_bound A)*> ^ D1) as non empty increasing FinSequence of REAL by A2, Lm8;

          

           A10: ( len MD1) = (( len <*( lower_bound A)*>) + ( len D1)) by FINSEQ_1: 22;

          (( len <*( lower_bound A)*>) + 1) <= (( len <*( lower_bound A)*>) + ( len D1)) by A7, XREAL_1: 6;

          

          then (MD1 . ( len MD1)) = (D1 . ((( len <*( lower_bound A)*>) + ( len D1)) - ( len <*( lower_bound A)*>))) by A10, FINSEQ_1: 23

          .= (D1 . ( len D1));

          then

           A11: (MD1 . ( len MD1)) = ( upper_bound A) by INTEGRA1:def 2;

          for y be Element of REAL holds y in ( rng MD1) implies y in A

          proof

            let y be Element of REAL ;

            assume y in ( rng MD1);

            then

             A12: y in (( rng <*( lower_bound A)*>) \/ ( rng D1)) by FINSEQ_1: 31;

            per cases by A12, XBOOLE_0:def 3;

              suppose y in ( rng <*( lower_bound A)*>);

              then y in {( lower_bound A)} by FINSEQ_1: 38;

              then

               A13: y = ( lower_bound A) by TARSKI:def 1;

              ex a, b st a <= b & a = ( lower_bound A) & b = ( upper_bound A) by SEQ_4: 11;

              hence thesis by A13, INTEGRA2: 1;

            end;

              suppose

               A14: y in ( rng D1);

              ( rng D1) c= A by INTEGRA1:def 2;

              hence thesis by A14;

            end;

          end;

          then ( rng MD1) c= A;

          then

          reconsider MD1 as Division of A by A11, INTEGRA1:def 2;

          

           A15: ( len MD1) = (( len <*( lower_bound A)*>) + ( len D1)) by FINSEQ_1: 22

          .= (1 + ( len D1)) by FINSEQ_1: 39;

          

           A16: ( vol A) >= 0 by INTEGRA1: 9;

          (D1 . 1) = ( upper_bound A) by A9, INTEGRA1:def 2;

          then ((D1 . 1) - ( lower_bound A)) > 0 by A2, A16, INTEGRA1:def 5;

          then

           A17: ( lower_bound A) < (D1 . 1) by XREAL_1: 47;

          ( lower_volume (f,D1)) = (( lower_volume (f,MD1)) /^ 1) by Lm10;

          then ( lower_volume (f,MD1)) = ( <*(( lower_volume (f,MD1)) /. 1)*> ^ ( lower_volume (f,D1))) by FINSEQ_5: 29;

          then

           A18: ( Sum ( lower_volume (f,MD1))) = ((( lower_volume (f,MD1)) /. 1) + ( Sum ( lower_volume (f,D1)))) by RVSUM_1: 76;

          

           A19: ( len D1) in ( dom D1) by FINSEQ_5: 6;

          

           A20: (1 + ( len D1)) >= (1 + 1) by A7, XREAL_1: 6;

          then

           A21: ( len MD1) <> 1 by A15;

          

           A22: ( len MD1) in ( dom MD1) by FINSEQ_5: 6;

          (( len MD1) - 1) = ( len D1) by A15;

          

          then ( lower_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len D1)) by A22, A21, INTEGRA1:def 4

          .= ( lower_bound A) by A9, FINSEQ_1: 41;

          then

           A23: ( lower_bound ( divset (D1,( len D1)))) = ( lower_bound ( divset (MD1,( len MD1)))) by A9, A19, INTEGRA1:def 4;

          set MD2 = ( <*( lower_bound A)*> ^ D2);

          ( rng MD1) <> {} ;

          then

           A24: 1 in ( dom MD1) by FINSEQ_3: 32;

          then

           A25: (( lower_volume (f,MD1)) . 1) = (( lower_bound ( rng (f | ( divset (MD1,1))))) * ( vol ( divset (MD1,1)))) by INTEGRA1:def 7;

          1 in ( Seg ( len MD1)) by A24, FINSEQ_1:def 3;

          then 1 in ( Seg ( len ( lower_volume (f,MD1)))) by INTEGRA1:def 7;

          then

           A26: 1 in ( dom ( lower_volume (f,MD1))) by FINSEQ_1:def 3;

          ( rng D2) <> {} ;

          then

           A27: 1 in ( dom D2) by FINSEQ_3: 32;

          then 1 <= ( len D2) by FINSEQ_3: 25;

          then

           A28: (( len <*( lower_bound A)*>) + 1) <= (( len <*( lower_bound A)*>) + ( len D2)) by XREAL_1: 6;

          

           A29: (D2 . 1) in ( rng D2) by A27, FUNCT_1:def 3;

          ( lower_bound A) < (D2 . 1)

          proof

            per cases by A4, A29, XBOOLE_0:def 3;

              suppose

               A30: (D2 . 1) in ( rng D1);

              ( rng D1) <> {} ;

              then

               A31: 1 in ( dom D1) by FINSEQ_3: 32;

              consider k such that

               A32: k in ( dom D1) and

               A33: (D1 . k) = (D2 . 1) by A30, PARTFUN1: 3;

              1 <= k by A32, FINSEQ_3: 25;

              then (D1 . 1) <= (D2 . 1) by A32, A33, A31, SEQ_4: 137;

              hence thesis by A17, XXREAL_0: 2;

            end;

              suppose (D2 . 1) in {x};

              hence thesis by A6, TARSKI:def 1;

            end;

          end;

          then

          reconsider MD2 as non empty increasing FinSequence of REAL by Lm9;

          ( len MD2) = (( len <*( lower_bound A)*>) + ( len D2)) by FINSEQ_1: 22;

          

          then (MD2 . ( len MD2)) = (D2 . ((( len <*( lower_bound A)*>) + ( len D2)) - ( len <*( lower_bound A)*>))) by A28, FINSEQ_1: 23

          .= (D2 . ( len D2));

          then

           A34: (MD2 . ( len MD2)) = ( upper_bound A) by INTEGRA1:def 2;

          for y be Element of REAL holds y in ( rng MD2) implies y in A

          proof

            let y be Element of REAL ;

            assume y in ( rng MD2);

            then

             A35: y in (( rng <*( lower_bound A)*>) \/ ( rng D2)) by FINSEQ_1: 31;

            per cases by A35, XBOOLE_0:def 3;

              suppose y in ( rng <*( lower_bound A)*>);

              then y in {( lower_bound A)} by FINSEQ_1: 38;

              then

               A36: y = ( lower_bound A) by TARSKI:def 1;

              ex a, b st a <= b & a = ( lower_bound A) & b = ( upper_bound A) by SEQ_4: 11;

              hence thesis by A36, INTEGRA2: 1;

            end;

              suppose

               A37: y in ( rng D2);

              ( rng D2) c= A by INTEGRA1:def 2;

              hence thesis by A37;

            end;

          end;

          then ( rng MD2) c= A;

          then

          reconsider MD2 as Division of A by A34, INTEGRA1:def 2;

          

           A38: x <= ( upper_bound ( divset (D1,( len D1)))) by A1, INTEGRA2: 1;

          ( rng MD2) = (( rng D2) \/ ( rng <*( lower_bound A)*>)) by FINSEQ_1: 31

          .= ((( rng D1) \/ ( rng <*( lower_bound A)*>)) \/ {x}) by A4, XBOOLE_1: 4;

          then

           A39: ( rng MD2) = (( rng MD1) \/ {x}) by FINSEQ_1: 31;

          (MD1 . ( len MD1)) = (MD1 . (( len <*( lower_bound A)*>) + ( len D1))) by FINSEQ_1: 22

          .= (D1 . ( len D1)) by A19, FINSEQ_1:def 7;

          

          then

           A40: ( upper_bound ( divset (MD1,( len MD1)))) = (D1 . ( len D1)) by A22, A21, INTEGRA1:def 4

          .= ( upper_bound ( divset (D1,( len D1)))) by A9, A19, INTEGRA1:def 4;

          ( rng D1) c= ( rng D2) by A3, INTEGRA1:def 18;

          then (( rng D1) \/ ( rng <*( lower_bound A)*>)) c= (( rng D2) \/ ( rng <*( lower_bound A)*>)) by XBOOLE_1: 9;

          then ( rng MD1) c= (( rng D2) \/ ( rng <*( lower_bound A)*>)) by FINSEQ_1: 31;

          then

           A41: ( rng MD1) c= ( rng MD2) by FINSEQ_1: 31;

          ( len D1) <= ( len D2) by A3, INTEGRA1:def 18;

          then (( len D1) + ( len <*( lower_bound A)*>)) <= (( len D2) + ( len <*( lower_bound A)*>)) by XREAL_1: 6;

          then ( len MD1) <= (( len D2) + ( len <*( lower_bound A)*>)) by FINSEQ_1: 22;

          then ( len MD1) <= ( len MD2) by FINSEQ_1: 22;

          then

           A42: MD1 <= MD2 by A41, INTEGRA1:def 18;

          ( lower_bound ( divset (D1,( len D1)))) <= x by A1, INTEGRA2: 1;

          then x in ( divset (MD1,( len MD1))) by A38, A23, A40, INTEGRA2: 1;

          then

           A43: (( Sum ( lower_volume (f,MD2))) - ( Sum ( lower_volume (f,MD1)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) by A5, A15, A20, A42, A39, Th10;

          ( rng MD2) <> {} ;

          then

           A44: 1 in ( dom MD2) by FINSEQ_3: 32;

          then

           A45: (( lower_volume (f,MD2)) . 1) = (( lower_bound ( rng (f | ( divset (MD2,1))))) * ( vol ( divset (MD2,1)))) by INTEGRA1:def 7;

          1 in ( Seg ( len MD2)) by A44, FINSEQ_1:def 3;

          then 1 in ( Seg ( len ( lower_volume (f,MD2)))) by INTEGRA1:def 7;

          then

           A46: 1 in ( dom ( lower_volume (f,MD2))) by FINSEQ_1:def 3;

          ( vol ( divset (MD2,1))) = 0 by Lm11;

          then

           A47: (( lower_volume (f,MD2)) /. 1) = 0 by A45, A46, PARTFUN1:def 6;

          ( lower_volume (f,D2)) = (( lower_volume (f,MD2)) /^ 1) by Lm10;

          then ( lower_volume (f,MD2)) = ( <*(( lower_volume (f,MD2)) /. 1)*> ^ ( lower_volume (f,D2))) by FINSEQ_5: 29;

          then

           A48: ( Sum ( lower_volume (f,MD2))) = ((( lower_volume (f,MD2)) /. 1) + ( Sum ( lower_volume (f,D2)))) by RVSUM_1: 76;

          ( vol ( divset (MD1,1))) = 0 by Lm11;

          then (( lower_volume (f,MD1)) /. 1) = 0 by A25, A26, PARTFUN1:def 6;

          hence thesis by A43, A18, A48, A47, Lm12;

        end;

          suppose ( len D1) >= 2;

          hence thesis by A1, A3, A4, A5, Th10;

        end;

      end;

      hence thesis;

    end;

    theorem :: INTEGRA3:15

    

     Th14: x in ( divset (D1,( len D1))) & ( vol A) <> 0 & D1 <= D2 & ( rng D2) = (( rng D1) \/ {x}) & (f | A) is bounded & x > ( lower_bound A) implies (( Sum ( upper_volume (f,D1))) - ( Sum ( upper_volume (f,D2)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))

    proof

      assume that

       A1: x in ( divset (D1,( len D1))) and

       A2: ( vol A) <> 0 and

       A3: D1 <= D2 and

       A4: ( rng D2) = (( rng D1) \/ {x}) and

       A5: (f | A) is bounded and

       A6: x > ( lower_bound A);

      ( len D1) in ( Seg ( len D1)) by FINSEQ_1: 3;

      then

       A7: 1 <= ( len D1) by FINSEQ_1: 1;

      then ( len D1) = 1 or ( len D1) > 1 by XXREAL_0: 1;

      then

       A8: ( len D1) = 1 or ( len D1) >= (1 + 1) by NAT_1: 13;

      now

        per cases by A8;

          suppose

           A9: ( len D1) = 1;

          then

          reconsider MD1 = ( <*( lower_bound A)*> ^ D1) as non empty increasing FinSequence of REAL by A2, Lm8;

          

           A10: ( len MD1) = (( len <*( lower_bound A)*>) + ( len D1)) by FINSEQ_1: 22;

          (( len <*( lower_bound A)*>) + 1) <= (( len <*( lower_bound A)*>) + ( len D1)) by A7, XREAL_1: 6;

          

          then (MD1 . ( len MD1)) = (D1 . ((( len <*( lower_bound A)*>) + ( len D1)) - ( len <*( lower_bound A)*>))) by A10, FINSEQ_1: 23

          .= (D1 . ( len D1));

          then

           A11: (MD1 . ( len MD1)) = ( upper_bound A) by INTEGRA1:def 2;

          for y be Element of REAL holds y in ( rng MD1) implies y in A

          proof

            let y be Element of REAL ;

            assume y in ( rng MD1);

            then

             A12: y in (( rng <*( lower_bound A)*>) \/ ( rng D1)) by FINSEQ_1: 31;

            per cases by A12, XBOOLE_0:def 3;

              suppose y in ( rng <*( lower_bound A)*>);

              then y in {( lower_bound A)} by FINSEQ_1: 38;

              then

               A13: y = ( lower_bound A) by TARSKI:def 1;

              ex a, b st a <= b & a = ( lower_bound A) & b = ( upper_bound A) by SEQ_4: 11;

              hence thesis by A13, INTEGRA2: 1;

            end;

              suppose

               A14: y in ( rng D1);

              ( rng D1) c= A by INTEGRA1:def 2;

              hence thesis by A14;

            end;

          end;

          then ( rng MD1) c= A;

          then

          reconsider MD1 as Division of A by A11, INTEGRA1:def 2;

          

           A15: ( len MD1) = (( len <*( lower_bound A)*>) + ( len D1)) by FINSEQ_1: 22

          .= (1 + ( len D1)) by FINSEQ_1: 39;

          

           A16: ( vol A) >= 0 by INTEGRA1: 9;

          (D1 . 1) = ( upper_bound A) by A9, INTEGRA1:def 2;

          then ((D1 . 1) - ( lower_bound A)) > 0 by A2, A16, INTEGRA1:def 5;

          then

           A17: ( lower_bound A) < (D1 . 1) by XREAL_1: 47;

          ( upper_volume (f,D1)) = (( upper_volume (f,MD1)) /^ 1) by Lm10;

          then ( upper_volume (f,MD1)) = ( <*(( upper_volume (f,MD1)) /. 1)*> ^ ( upper_volume (f,D1))) by FINSEQ_5: 29;

          then

           A18: ( Sum ( upper_volume (f,MD1))) = ((( upper_volume (f,MD1)) /. 1) + ( Sum ( upper_volume (f,D1)))) by RVSUM_1: 76;

          

           A19: ( len D1) in ( dom D1) by FINSEQ_5: 6;

          

           A20: (1 + ( len D1)) >= (1 + 1) by A7, XREAL_1: 6;

          then

           A21: ( len MD1) <> 1 by A15;

          

           A22: ( len MD1) in ( dom MD1) by FINSEQ_5: 6;

          (( len MD1) - 1) = ( len D1) by A15;

          

          then ( lower_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len D1)) by A22, A21, INTEGRA1:def 4

          .= ( lower_bound A) by A9, FINSEQ_1: 41;

          then

           A23: ( lower_bound ( divset (D1,( len D1)))) = ( lower_bound ( divset (MD1,( len MD1)))) by A9, A19, INTEGRA1:def 4;

          set MD2 = ( <*( lower_bound A)*> ^ D2);

          ( rng MD1) <> {} ;

          then

           A24: 1 in ( dom MD1) by FINSEQ_3: 32;

          then

           A25: (( upper_volume (f,MD1)) . 1) = (( upper_bound ( rng (f | ( divset (MD1,1))))) * ( vol ( divset (MD1,1)))) by INTEGRA1:def 6;

          1 in ( Seg ( len MD1)) by A24, FINSEQ_1:def 3;

          then 1 in ( Seg ( len ( upper_volume (f,MD1)))) by INTEGRA1:def 6;

          then

           A26: 1 in ( dom ( upper_volume (f,MD1))) by FINSEQ_1:def 3;

          ( rng D2) <> {} ;

          then

           A27: 1 in ( dom D2) by FINSEQ_3: 32;

          then 1 <= ( len D2) by FINSEQ_3: 25;

          then

           A28: (( len <*( lower_bound A)*>) + 1) <= (( len <*( lower_bound A)*>) + ( len D2)) by XREAL_1: 6;

          

           A29: (D2 . 1) in ( rng D2) by A27, FUNCT_1:def 3;

          ( lower_bound A) < (D2 . 1)

          proof

            per cases by A4, A29, XBOOLE_0:def 3;

              suppose

               A30: (D2 . 1) in ( rng D1);

              ( rng D1) <> {} ;

              then

               A31: 1 in ( dom D1) by FINSEQ_3: 32;

              consider k such that

               A32: k in ( dom D1) and

               A33: (D1 . k) = (D2 . 1) by A30, PARTFUN1: 3;

              1 <= k by A32, FINSEQ_3: 25;

              then (D1 . 1) <= (D2 . 1) by A32, A33, A31, SEQ_4: 137;

              hence thesis by A17, XXREAL_0: 2;

            end;

              suppose (D2 . 1) in {x};

              hence thesis by A6, TARSKI:def 1;

            end;

          end;

          then

          reconsider MD2 as non empty increasing FinSequence of REAL by Lm9;

          ( len MD2) = (( len <*( lower_bound A)*>) + ( len D2)) by FINSEQ_1: 22;

          

          then (MD2 . ( len MD2)) = (D2 . ((( len <*( lower_bound A)*>) + ( len D2)) - ( len <*( lower_bound A)*>))) by A28, FINSEQ_1: 23

          .= (D2 . ( len D2));

          then

           A34: (MD2 . ( len MD2)) = ( upper_bound A) by INTEGRA1:def 2;

          for y be Element of REAL holds y in ( rng MD2) implies y in A

          proof

            let y be Element of REAL ;

            assume y in ( rng MD2);

            then

             A35: y in (( rng <*( lower_bound A)*>) \/ ( rng D2)) by FINSEQ_1: 31;

            per cases by A35, XBOOLE_0:def 3;

              suppose y in ( rng <*( lower_bound A)*>);

              then y in {( lower_bound A)} by FINSEQ_1: 38;

              then

               A36: y = ( lower_bound A) by TARSKI:def 1;

              ex a, b st a <= b & a = ( lower_bound A) & b = ( upper_bound A) by SEQ_4: 11;

              hence thesis by A36, INTEGRA2: 1;

            end;

              suppose

               A37: y in ( rng D2);

              ( rng D2) c= A by INTEGRA1:def 2;

              hence thesis by A37;

            end;

          end;

          then ( rng MD2) c= A;

          then

          reconsider MD2 as Division of A by A34, INTEGRA1:def 2;

          

           A38: x <= ( upper_bound ( divset (D1,( len D1)))) by A1, INTEGRA2: 1;

          ( rng MD2) = (( rng D2) \/ ( rng <*( lower_bound A)*>)) by FINSEQ_1: 31

          .= ((( rng D1) \/ ( rng <*( lower_bound A)*>)) \/ {x}) by A4, XBOOLE_1: 4;

          then

           A39: ( rng MD2) = (( rng MD1) \/ {x}) by FINSEQ_1: 31;

          (MD1 . ( len MD1)) = (MD1 . (( len <*( lower_bound A)*>) + ( len D1))) by FINSEQ_1: 22

          .= (D1 . ( len D1)) by A19, FINSEQ_1:def 7;

          

          then

           A40: ( upper_bound ( divset (MD1,( len MD1)))) = (D1 . ( len D1)) by A22, A21, INTEGRA1:def 4

          .= ( upper_bound ( divset (D1,( len D1)))) by A9, A19, INTEGRA1:def 4;

          ( rng D1) c= ( rng D2) by A3, INTEGRA1:def 18;

          then (( rng D1) \/ ( rng <*( lower_bound A)*>)) c= (( rng D2) \/ ( rng <*( lower_bound A)*>)) by XBOOLE_1: 9;

          then ( rng MD1) c= (( rng D2) \/ ( rng <*( lower_bound A)*>)) by FINSEQ_1: 31;

          then

           A41: ( rng MD1) c= ( rng MD2) by FINSEQ_1: 31;

          ( len D1) <= ( len D2) by A3, INTEGRA1:def 18;

          then (( len D1) + ( len <*( lower_bound A)*>)) <= (( len D2) + ( len <*( lower_bound A)*>)) by XREAL_1: 6;

          then ( len MD1) <= (( len D2) + ( len <*( lower_bound A)*>)) by FINSEQ_1: 22;

          then ( len MD1) <= ( len MD2) by FINSEQ_1: 22;

          then

           A42: MD1 <= MD2 by A41, INTEGRA1:def 18;

          ( lower_bound ( divset (D1,( len D1)))) <= x by A1, INTEGRA2: 1;

          then x in ( divset (MD1,( len MD1))) by A38, A23, A40, INTEGRA2: 1;

          then

           A43: (( Sum ( upper_volume (f,MD1))) - ( Sum ( upper_volume (f,MD2)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) by A5, A15, A20, A42, A39, Th11;

          ( rng MD2) <> {} ;

          then

           A44: 1 in ( dom MD2) by FINSEQ_3: 32;

          then

           A45: (( upper_volume (f,MD2)) . 1) = (( upper_bound ( rng (f | ( divset (MD2,1))))) * ( vol ( divset (MD2,1)))) by INTEGRA1:def 6;

          1 in ( Seg ( len MD2)) by A44, FINSEQ_1:def 3;

          then 1 in ( Seg ( len ( upper_volume (f,MD2)))) by INTEGRA1:def 6;

          then

           A46: 1 in ( dom ( upper_volume (f,MD2))) by FINSEQ_1:def 3;

          ( vol ( divset (MD2,1))) = 0 by Lm11;

          then

           A47: (( upper_volume (f,MD2)) /. 1) = 0 by A45, A46, PARTFUN1:def 6;

          ( upper_volume (f,D2)) = (( upper_volume (f,MD2)) /^ 1) by Lm10;

          then ( upper_volume (f,MD2)) = ( <*(( upper_volume (f,MD2)) /. 1)*> ^ ( upper_volume (f,D2))) by FINSEQ_5: 29;

          then

           A48: ( Sum ( upper_volume (f,MD2))) = ((( upper_volume (f,MD2)) /. 1) + ( Sum ( upper_volume (f,D2)))) by RVSUM_1: 76;

          ( vol ( divset (MD1,1))) = 0 by Lm11;

          then (( upper_volume (f,MD1)) /. 1) = 0 by A25, A26, PARTFUN1:def 6;

          hence thesis by A43, A18, A48, A47, Lm12;

        end;

          suppose ( len D1) >= 2;

          hence thesis by A1, A3, A4, A5, Th11;

        end;

      end;

      hence thesis;

    end;

    theorem :: INTEGRA3:16

    

     Th15: i in ( dom D1) & j in ( dom D1) & i <= j & D1 <= D2 & r < (( mid (D2,( indx (D2,D1,i)),( indx (D2,D1,j)))) . 1) implies ex B be non empty closed_interval Subset of REAL , MD1,MD2 be Division of B st r = ( lower_bound B) & ( upper_bound B) = (MD2 . ( len MD2)) & ( upper_bound B) = (MD1 . ( len MD1)) & MD1 <= MD2 & MD1 = ( mid (D1,i,j)) & MD2 = ( mid (D2,( indx (D2,D1,i)),( indx (D2,D1,j))))

    proof

      set MD1 = ( mid (D1,i,j));

      set MD2 = ( mid (D2,( indx (D2,D1,i)),( indx (D2,D1,j))));

      assume

       A1: i in ( dom D1);

      then

       A2: 1 <= i by FINSEQ_3: 25;

      assume

       A3: j in ( dom D1);

      assume

       A4: i <= j;

      then (j - i) >= 0 by XREAL_1: 48;

      then

       A5: ((j - i) + 1) >= ( 0 + 1) by XREAL_1: 6;

      

       A6: j <= ( len D1) by A3, FINSEQ_3: 25;

      

      then

       A7: (MD1 . 1) = (D1 . ((1 + i) - 1)) by A4, A5, A2, FINSEQ_6: 122

      .= (D1 . i);

      assume

       A8: D1 <= D2;

      then

       A9: (D2 . ( indx (D2,D1,i))) = (D1 . i) by A1, INTEGRA1:def 19;

      

       A10: (D2 . ( indx (D2,D1,j))) = (D1 . j) by A3, A8, INTEGRA1:def 19;

      

       A11: ( indx (D2,D1,i)) in ( dom D2) by A1, A8, INTEGRA1:def 19;

      then

       A12: 1 <= ( indx (D2,D1,i)) by FINSEQ_3: 25;

      

       A13: ( indx (D2,D1,j)) in ( dom D2) by A3, A8, INTEGRA1:def 19;

      then

       A14: ( indx (D2,D1,j)) <= ( len D2) by FINSEQ_3: 25;

      (D1 . i) <= (D1 . j) by A1, A3, A4, SEQ_4: 137;

      then

       A15: ( indx (D2,D1,i)) <= ( indx (D2,D1,j)) by A11, A9, A13, A10, SEQM_3:def 1;

      assume

       A16: r < (( mid (D2,( indx (D2,D1,i)),( indx (D2,D1,j)))) . 1);

      then

      consider B be non empty closed_interval Subset of REAL such that

       A17: r = ( lower_bound B) and

       A18: ( upper_bound B) = (MD2 . ( len MD2)) and

       A19: MD2 is Division of B by A11, A13, A15, Th12;

      

       A20: ( len MD2) = ((( indx (D2,D1,j)) - ( indx (D2,D1,i))) + 1) by A11, A13, A15, INTEGRA1: 58;

      reconsider MD2 as Division of B by A19;

      (( indx (D2,D1,j)) - ( indx (D2,D1,i))) >= 0 by A15, XREAL_1: 48;

      then

       A21: ((( indx (D2,D1,j)) - ( indx (D2,D1,i))) + 1) >= ( 0 + 1) by XREAL_1: 6;

      

      then

       A22: (MD2 . ( len MD2)) = (D2 . ((((( indx (D2,D1,j)) - ( indx (D2,D1,i))) + 1) - 1) + ( indx (D2,D1,i)))) by A15, A20, A12, A14, FINSEQ_6: 122

      .= (D1 . j) by A3, A8, INTEGRA1:def 19;

      (MD2 . 1) = (D2 . ((1 + ( indx (D2,D1,i))) - 1)) by A15, A21, A12, A14, FINSEQ_6: 122

      .= (D1 . i) by A1, A8, INTEGRA1:def 19;

      then

      consider C be non empty closed_interval Subset of REAL such that

       A23: r = ( lower_bound C) and

       A24: ( upper_bound C) = (MD1 . ( len MD1)) and

       A25: MD1 is Division of C by A1, A3, A4, A16, A7, Th12;

      ( len MD1) = ((j - i) + 1) by A1, A3, A4, INTEGRA1: 58;

      

      then

       A26: (MD1 . ( len MD1)) = (D1 . ((((j - i) + 1) - 1) + i)) by A4, A5, A2, A6, FINSEQ_6: 122

      .= (D1 . j);

      

       A27: B = [.( lower_bound B), ( upper_bound B).] by INTEGRA1: 4

      .= C by A17, A18, A23, A24, A26, A22, INTEGRA1: 4;

      then

      reconsider MD1 as Division of B by A25;

      

       A28: ( rng MD1) c= ( rng MD2)

      proof

        let x1 be object;

        

         A29: ( rng MD1) c= ( rng D1) by FINSEQ_6: 119;

        assume

         A30: x1 in ( rng MD1);

        then

        consider k1 be Element of NAT such that

         A31: k1 in ( dom MD1) and

         A32: (MD1 . k1) = x1 by PARTFUN1: 3;

        ( rng D1) c= ( rng D2) by A8, INTEGRA1:def 18;

        then ( rng MD1) c= ( rng D2) by A29;

        then

        consider k2 be Element of NAT such that

         A33: k2 in ( dom D2) and

         A34: (D2 . k2) = x1 by A30, PARTFUN1: 3;

        

         A35: k1 <= ( len MD1) by A31, FINSEQ_3: 25;

        

         A36: 1 <= k1 by A31, FINSEQ_3: 25;

        then 1 <= ( len MD1) by A35, XXREAL_0: 2;

        then 1 in ( dom MD1) by FINSEQ_3: 25;

        then (MD1 . 1) <= (MD1 . k1) by A31, A36, SEQ_4: 137;

        then

         A37: ( indx (D2,D1,i)) <= k2 by A11, A9, A7, A33, A34, A32, SEQM_3:def 1;

        then

        consider k3 be Nat such that

         A38: (k2 + 1) = (( indx (D2,D1,i)) + k3) by NAT_1: 10, NAT_1: 12;

        ( len MD1) in ( dom MD1) by FINSEQ_5: 6;

        then (MD1 . k1) <= (MD1 . ( len MD1)) by A31, A35, SEQ_4: 137;

        then k2 <= ( indx (D2,D1,j)) by A13, A10, A26, A33, A34, A32, SEQM_3:def 1;

        then (k2 + 1) <= (( indx (D2,D1,j)) + 1) by XREAL_1: 6;

        then

         A39: ((k2 + 1) - ( indx (D2,D1,i))) <= ((( indx (D2,D1,j)) + 1) - ( indx (D2,D1,i))) by XREAL_1: 9;

        (( indx (D2,D1,i)) + 1) <= (k2 + 1) by A37, XREAL_1: 6;

        then

         A40: 1 <= ((k2 + 1) - ( indx (D2,D1,i))) by XREAL_1: 19;

        then

         A41: k3 in ( dom MD2) by A20, A39, A38, FINSEQ_3: 25;

        (MD2 . k3) = (D2 . ((k3 + ( indx (D2,D1,i))) - 1)) by A15, A12, A14, A40, A39, A38, FINSEQ_6: 122;

        hence thesis by A34, A38, A41, FUNCT_1:def 3;

      end;

      

       A42: ( card ( rng MD2)) = ( len MD2) by FINSEQ_4: 62;

      ( card ( rng MD1)) = ( len MD1) by FINSEQ_4: 62;

      then ( len MD1) <= ( len MD2) by A28, A42, NAT_1: 43;

      then MD1 <= MD2 by A28, INTEGRA1:def 18;

      hence thesis by A17, A18, A24, A27;

    end;

    theorem :: INTEGRA3:17

    

     Th16: x in ( rng D) implies (D . 1) <= x & x <= (D . ( len D))

    proof

      assume x in ( rng D);

      then

      consider i such that

       A1: i in ( dom D) and

       A2: x = (D . i) by PARTFUN1: 3;

      

       A3: i <= ( len D) by A1, FINSEQ_3: 25;

      

       A4: 1 <= i by A1, FINSEQ_3: 25;

      then

       A5: 1 <= ( len D) by A3, XXREAL_0: 2;

      then

       A6: ( len D) in ( dom D) by FINSEQ_3: 25;

      1 in ( dom D) by A5, FINSEQ_3: 25;

      hence thesis by A1, A2, A4, A3, A6, SEQ_4: 137;

    end;

    theorem :: INTEGRA3:18

    

     Th17: for p be FinSequence of REAL , i, j, k st p is increasing & i in ( dom p) & j in ( dom p) & k in ( dom p) & (p . i) <= (p . k) & (p . k) <= (p . j) holds (p . k) in ( rng ( mid (p,i,j)))

    proof

      let p be FinSequence of REAL ;

      let i, j, k;

      assume that

       A1: p is increasing and

       A2: i in ( dom p) and

       A3: j in ( dom p) and

       A4: k in ( dom p) and

       A5: (p . i) <= (p . k) and

       A6: (p . k) <= (p . j);

      

       A7: 1 <= i by A2, FINSEQ_3: 25;

      

       A8: 1 <= j by A3, FINSEQ_3: 25;

      

       A9: j <= ( len p) by A3, FINSEQ_3: 25;

      

       A10: i <= k by A1, A2, A4, A5, SEQM_3:def 1;

      then

      consider n be Nat such that

       A11: (k + 1) = (i + n) by NAT_1: 10, NAT_1: 12;

      

       A12: k <= j by A1, A3, A4, A6, SEQM_3:def 1;

      then (k - i) <= (j - i) by XREAL_1: 9;

      then

       A13: ((k - i) + 1) <= ((j - i) + 1) by XREAL_1: 6;

      (k - i) >= 0 by A10, XREAL_1: 48;

      then

       A14: ((k - i) + 1) >= ( 0 + 1) by XREAL_1: 6;

      

       A15: i <= j by A10, A12, XXREAL_0: 2;

      i <= ( len p) by A2, FINSEQ_3: 25;

      then ( len ( mid (p,i,j))) = ((j -' i) + 1) by A7, A8, A9, A15, FINSEQ_6: 118;

      then ( len ( mid (p,i,j))) = ((j - i) + 1) by A10, A12, XREAL_1: 233, XXREAL_0: 2;

      then

       A16: n in ( dom ( mid (p,i,j))) by A11, A14, A13, FINSEQ_3: 25;

      (( mid (p,i,j)) . n) = (p . ((n + i) - 1)) by A7, A9, A15, A11, A14, A13, FINSEQ_6: 122

      .= (p . k) by A11;

      hence thesis by A16, FUNCT_1:def 3;

    end;

    theorem :: INTEGRA3:19

    

     Th18: (f | A) is bounded & i in ( dom D) implies ( lower_bound ( rng (f | ( divset (D,i))))) <= ( upper_bound ( rng f))

    proof

      assume

       A1: (f | A) is bounded;

      assume i in ( dom D);

      then ( divset (D,i)) c= A by INTEGRA1: 8;

      hence thesis by A1, Lm4;

    end;

    theorem :: INTEGRA3:20

    

     Th19: (f | A) is bounded & i in ( dom D) implies ( upper_bound ( rng (f | ( divset (D,i))))) >= ( lower_bound ( rng f))

    proof

      assume

       A1: (f | A) is bounded;

      assume i in ( dom D);

      then ( divset (D,i)) c= A by INTEGRA1: 8;

      hence thesis by A1, Lm4;

    end;

    begin

    theorem :: INTEGRA3:21

    

     Th20: (f | A) is bounded implies for D, D1 holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D))) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D1)))

    proof

      assume

       A1: (f | A) is bounded;

      for D, D1 holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D))) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D1)))

      proof

        let D, D1;

        consider D2 such that

         A6: D <= D2 and

         A7: D1 <= D2 and

         A8: ( rng D2) = (( rng D1) \/ ( rng D)) by Th4;

        

         A9: (( lower_sum (f,D2)) - ( lower_sum (f,D1))) >= 0 by A1, A7, INTEGRA1: 46, XREAL_1: 48;

        (( lower_sum (f,D2)) - ( lower_sum (f,D))) >= 0 by A1, A6, INTEGRA1: 46, XREAL_1: 48;

        hence thesis by A6, A7, A8, A9;

      end;

      hence thesis;

    end;

    theorem :: INTEGRA3:22

    

     Th21: (f | A) is bounded implies for D, D1 st ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D)))) holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & (( lower_sum (f,D2)) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

    proof

      assume

       A1: (f | A) is bounded;

      then

       A5: for D, D1 holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D))) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D1))) by Th20;

      for D, D1 st ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D)))) holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & (( lower_sum (f,D2)) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

      proof

        let D, D1;

        assume

         A11: ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D))));

        ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & (( lower_sum (f,D2)) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

        proof

          consider D2 such that

           A12: D <= D2 and

           A13: D1 <= D2 and

           A14: ( rng D2) = (( rng D1) \/ ( rng D)) and 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D))) and 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D1))) by A5;

          (( lower_sum (f,D2)) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

          proof

            deffunc LVf( Division of A) = ( lower_volume (f,$1));

            deffunc PLf( Division of A, Nat) = (( PartSums ( lower_volume (f,$1))) . $2);

            

             A15: ( len D2) in ( dom D2) by FINSEQ_5: 6;

            

             A16: for i st i in ( dom D) holds ex j st j in ( dom D1) & (D . i) in ( divset (D1,j)) & ( PLf(D2,indx) - PLf(D1,j)) <= ((i * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

            proof

              defpred P[ non zero Nat] means $1 in ( dom D) implies ex j st j in ( dom D1) & (D . $1) in ( divset (D1,j)) & ( PLf(D2,indx) - PLf(D1,j)) <= (($1 * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1));

              let i;

              assume

               A17: i in ( dom D);

              then

               A18: i in ( Seg ( len D)) by FINSEQ_1:def 3;

              

               A19: for i, j st i in ( dom D) & j in ( dom D1) & (D . i) in ( divset (D1,j)) holds j >= 2

              proof

                let i, j;

                assume

                 A20: i in ( dom D);

                assume that

                 A21: j in ( dom D1) and

                 A22: (D . i) in ( divset (D1,j));

                assume j < 2;

                then j < (1 + 1);

                then

                 A23: j <= 1 by NAT_1: 13;

                j in ( Seg ( len D1)) by A21, FINSEQ_1:def 3;

                then j >= 1 by FINSEQ_1: 1;

                then j = 1 by A23, XXREAL_0: 1;

                then

                 A24: ( lower_bound ( divset (D1,j))) = ( lower_bound A) by A21, INTEGRA1:def 4;

                

                 A25: (D . i) <= ( upper_bound ( divset (D1,j))) by A22, INTEGRA2: 1;

                ( delta D1) >= ( min ( rng ( upper_volume (( chi (A,A)),D))))

                proof

                  per cases ;

                    suppose

                     A26: i = 1;

                    ( len D) in ( Seg ( len D)) by FINSEQ_1: 3;

                    then 1 <= ( len D) by FINSEQ_1: 1;

                    then

                     A27: 1 in ( Seg ( len D)) by FINSEQ_1: 1;

                    then

                     A28: 1 in ( dom D) by FINSEQ_1:def 3;

                    then

                     A29: ( lower_bound ( divset (D,1))) = ( lower_bound A) by INTEGRA1:def 4;

                    1 in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by A27, INTEGRA1:def 6;

                    then

                     A30: 1 in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

                    ( vol ( divset (D,1))) = (( upper_volume (( chi (A,A)),D)) . 1) by A28, INTEGRA1: 20;

                    then ( vol ( divset (D,1))) in ( rng ( upper_volume (( chi (A,A)),D))) by A30, FUNCT_1:def 3;

                    then

                     A31: ( vol ( divset (D,1))) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by XXREAL_2:def 7;

                    

                     A32: ( upper_bound ( divset (D,1))) = (D . 1) by A28, INTEGRA1:def 4;

                    (( upper_bound ( divset (D1,j))) - ( lower_bound A)) >= ((D . 1) - ( lower_bound A)) by A25, A26, XREAL_1: 9;

                    then ( vol ( divset (D1,j))) >= (( upper_bound ( divset (D,1))) - ( lower_bound ( divset (D,1)))) by A24, A29, A32, INTEGRA1:def 5;

                    then

                     A33: ( vol ( divset (D1,j))) >= ( vol ( divset (D,1))) by INTEGRA1:def 5;

                    ( vol ( divset (D1,j))) <= ( delta D1) by A21, Lm5;

                    then ( delta D1) >= ( vol ( divset (D,1))) by A33, XXREAL_0: 2;

                    hence thesis by A31, XXREAL_0: 2;

                  end;

                    suppose

                     A34: i <> 1;

                    then (D . (i - 1)) in A by A20, INTEGRA1: 7;

                    then

                     A35: ( lower_bound A) <= (D . (i - 1)) by INTEGRA2: 1;

                    ( lower_bound ( divset (D,i))) = (D . (i - 1)) by A20, A34, INTEGRA1:def 4;

                    then

                     A36: (( upper_bound ( divset (D,i))) - ( lower_bound A)) >= (( upper_bound ( divset (D,i))) - ( lower_bound ( divset (D,i)))) by A35, XREAL_1: 10;

                    ( upper_bound ( divset (D,i))) = (D . i) by A20, A34, INTEGRA1:def 4;

                    then (( upper_bound ( divset (D1,j))) - ( lower_bound ( divset (D1,j)))) >= (( upper_bound ( divset (D,i))) - ( lower_bound A)) by A25, A24, XREAL_1: 9;

                    then (( upper_bound ( divset (D1,j))) - ( lower_bound ( divset (D1,j)))) >= (( upper_bound ( divset (D,i))) - ( lower_bound ( divset (D,i)))) by A36, XXREAL_0: 2;

                    then ( vol ( divset (D1,j))) >= (( upper_bound ( divset (D,i))) - ( lower_bound ( divset (D,i)))) by INTEGRA1:def 5;

                    then

                     A37: ( vol ( divset (D1,j))) >= ( vol ( divset (D,i))) by INTEGRA1:def 5;

                    i in ( Seg ( len D)) by A20, FINSEQ_1:def 3;

                    then i in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by INTEGRA1:def 6;

                    then

                     A38: i in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

                    ( vol ( divset (D,i))) = (( upper_volume (( chi (A,A)),D)) . i) by A20, INTEGRA1: 20;

                    then ( vol ( divset (D,i))) in ( rng ( upper_volume (( chi (A,A)),D))) by A38, FUNCT_1:def 3;

                    then

                     A39: ( vol ( divset (D,i))) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by XXREAL_2:def 7;

                    ( vol ( divset (D1,j))) <= ( delta D1) by A21, Lm5;

                    then ( delta D1) >= ( vol ( divset (D,i))) by A37, XXREAL_0: 2;

                    hence thesis by A39, XXREAL_0: 2;

                  end;

                end;

                hence contradiction by A11;

              end;

              

               A40: P[1]

              proof

                ( len D) in ( Seg ( len D)) by FINSEQ_1: 3;

                then 1 <= ( len D) by FINSEQ_1: 1;

                then

                 A41: 1 in ( dom D) by FINSEQ_3: 25;

                then

                consider j such that

                 A42: j in ( dom D1) and

                 A43: (D . 1) in ( divset (D1,j)) by Th3, INTEGRA1: 6;

                ( PLf(D2,indx) - PLf(D1,j)) <= ((1 * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

                proof

                  

                   A44: j <> 1 by A19, A41, A42, A43;

                  then

                  reconsider j1 = (j - 1) as Element of NAT by A42, INTEGRA1: 7;

                  

                   A45: j1 in ( dom D1) by A42, A44, INTEGRA1: 7;

                  then j1 in ( Seg ( len D1)) by FINSEQ_1:def 3;

                  then j1 in ( Seg ( len ( lower_volume (f,D1)))) by INTEGRA1:def 7;

                  then

                   A46: j1 in ( dom ( lower_volume (f,D1))) by FINSEQ_1:def 3;

                  

                   A47: (j - 1) in ( dom D1) by A42, A44, INTEGRA1: 7;

                  then

                   A48: ( indx (D2,D1,j1)) in ( dom D2) by A13, INTEGRA1:def 19;

                  then

                   A49: ( indx (D2,D1,j1)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                  then

                   A50: 1 <= ( indx (D2,D1,j1)) by FINSEQ_1: 1;

                  then ( mid (D2,1,( indx (D2,D1,j1)))) is increasing by A48, INTEGRA1: 35;

                  then

                   A51: (D2 | ( indx (D2,D1,j1))) is increasing by A50, FINSEQ_6: 116;

                  j < (j + 1) by NAT_1: 13;

                  then j1 < j by XREAL_1: 19;

                  then

                   A52: ( indx (D2,D1,j1)) < ( indx (D2,D1,j)) by A13, A42, A45, Th8;

                  then

                   A53: (( indx (D2,D1,j1)) + 1) <= ( indx (D2,D1,j)) by NAT_1: 13;

                  

                   A54: (( Sum ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) - ( Sum ( mid (( lower_volume (f,D1)),j,j)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))

                  proof

                    

                     A55: (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) <= 2

                    proof

                      reconsider ID1 = (( indx (D2,D1,j1)) + 1) as Element of NAT ;

                      reconsider ID2 = (ID1 + 1) as Element of NAT ;

                      assume (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) > 2;

                      then

                       A56: (( indx (D2,D1,j1)) + (1 + 1)) < ( indx (D2,D1,j)) by XREAL_1: 20;

                      

                       A57: ID1 < ID2 by NAT_1: 13;

                      then ( indx (D2,D1,j1)) <= ID2 by NAT_1: 13;

                      then

                       A58: 1 <= ID2 by A50, XXREAL_0: 2;

                      

                       A59: ( indx (D2,D1,j)) in ( dom D2) by A13, A42, INTEGRA1:def 19;

                      then

                       A60: ( indx (D2,D1,j)) <= ( len D2) by FINSEQ_3: 25;

                      then ID2 <= ( len D2) by A56, XXREAL_0: 2;

                      then ID2 in ( Seg ( len D2)) by A58, FINSEQ_1: 1;

                      then

                       A61: ID2 in ( dom D2) by FINSEQ_1:def 3;

                      then

                       A62: (D2 . ID2) < (D2 . ( indx (D2,D1,j))) by A56, A59, SEQM_3:def 1;

                      

                       A63: 1 <= ID1 by A50, NAT_1: 13;

                      

                       A64: (D1 . j) = (D2 . ( indx (D2,D1,j))) by A13, A42, INTEGRA1:def 19;

                      ID1 <= ( indx (D2,D1,j)) by A56, A57, XXREAL_0: 2;

                      then ID1 <= ( len D2) by A60, XXREAL_0: 2;

                      then ID1 in ( Seg ( len D2)) by A63, FINSEQ_1: 1;

                      then

                       A65: ID1 in ( dom D2) by FINSEQ_1:def 3;

                      then

                       A66: (D2 . ID1) < (D2 . ID2) by A57, A61, SEQM_3:def 1;

                      ( indx (D2,D1,j1)) < ID1 by NAT_1: 13;

                      then

                       A67: (D2 . ( indx (D2,D1,j1))) < (D2 . ID1) by A48, A65, SEQM_3:def 1;

                      

                       A68: (D1 . j1) = (D2 . ( indx (D2,D1,j1))) by A13, A45, INTEGRA1:def 19;

                      

                       A69: not (D2 . ID1) in ( rng D1) & not (D2 . ID2) in ( rng D1)

                      proof

                        assume

                         A70: (D2 . ID1) in ( rng D1) or (D2 . ID2) in ( rng D1);

                        per cases by A70;

                          suppose (D2 . ID1) in ( rng D1);

                          then

                          consider n such that

                           A71: n in ( dom D1) and

                           A72: (D1 . n) = (D2 . ID1) by PARTFUN1: 3;

                          j1 < n by A45, A67, A68, A71, A72, SEQ_4: 137;

                          then

                           A73: j < (n + 1) by XREAL_1: 19;

                          (D2 . ID1) < (D2 . ( indx (D2,D1,j))) by A66, A62, XXREAL_0: 2;

                          then n < j by A42, A64, A71, A72, SEQ_4: 137;

                          hence contradiction by A73, NAT_1: 13;

                        end;

                          suppose (D2 . ID2) in ( rng D1);

                          then

                          consider n such that

                           A74: n in ( dom D1) and

                           A75: (D1 . n) = (D2 . ID2) by PARTFUN1: 3;

                          (D2 . ( indx (D2,D1,j1))) < (D2 . ID2) by A67, A66, XXREAL_0: 2;

                          then j1 < n by A45, A68, A74, A75, SEQ_4: 137;

                          then

                           A76: j < (n + 1) by XREAL_1: 19;

                          n < j by A42, A62, A64, A74, A75, SEQ_4: 137;

                          hence contradiction by A76, NAT_1: 13;

                        end;

                      end;

                      ( upper_bound ( divset (D1,j))) = (D1 . j) by A42, A44, INTEGRA1:def 4;

                      then

                       A77: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A13, A42, INTEGRA1:def 19;

                      ( lower_bound ( divset (D1,j))) = (D1 . j1) by A42, A44, INTEGRA1:def 4;

                      then

                       A78: ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A13, A45, INTEGRA1:def 19;

                      (D2 . ID2) in (( rng D) \/ ( rng D1)) by A14, A61, FUNCT_1:def 3;

                      then

                       A79: (D2 . ID2) in ( rng D) by A69, XBOOLE_0:def 3;

                      (D2 . ID1) in (( rng D) \/ ( rng D1)) by A14, A65, FUNCT_1:def 3;

                      then

                       A80: (D2 . ID1) in ( rng D) by A69, XBOOLE_0:def 3;

                      (D2 . ( indx (D2,D1,j1))) <= (D2 . ID2) by A67, A66, XXREAL_0: 2;

                      then (D2 . ID2) in ( divset (D1,j)) by A62, A78, A77, INTEGRA2: 1;

                      then

                       A81: (D2 . ID2) in (( rng D) /\ ( divset (D1,j))) by A79, XBOOLE_0:def 4;

                      (D2 . ID1) <= (D2 . ( indx (D2,D1,j))) by A66, A62, XXREAL_0: 2;

                      then (D2 . ID1) in ( divset (D1,j)) by A67, A78, A77, INTEGRA2: 1;

                      then (D2 . ID1) in (( rng D) /\ ( divset (D1,j))) by A80, XBOOLE_0:def 4;

                      hence contradiction by A11, A42, A57, A65, A61, A81, Th5, SEQ_4: 138;

                    end;

                    

                     A82: 1 <= (( indx (D2,D1,j1)) + 1) by A50, NAT_1: 13;

                    j <= ( len D1) by A42, FINSEQ_3: 25;

                    then

                     A83: j <= ( len ( lower_volume (f,D1))) by INTEGRA1:def 7;

                    

                     A84: 1 <= j by A42, FINSEQ_3: 25;

                    then

                     A85: (( mid (( lower_volume (f,D1)),j,j)) . 1) = (( lower_volume (f,D1)) . j) by A83, FINSEQ_6: 118;

                    reconsider lv = (( lower_volume (f,D1)) . j) as Element of REAL by XREAL_0:def 1;

                    ((j -' j) + 1) = 1 by Lm1;

                    then ( len ( mid (( lower_volume (f,D1)),j,j))) = 1 by A84, A83, FINSEQ_6: 118;

                    then ( mid (( lower_volume (f,D1)),j,j)) = <*lv*> by A85, FINSEQ_1: 40;

                    then

                     A86: ( Sum ( mid (( lower_volume (f,D1)),j,j))) = (( lower_volume (f,D1)) . j) by FINSOP_1: 11;

                    ( indx (D2,D1,j)) in ( dom D2) by A13, A42, INTEGRA1:def 19;

                    then

                     A87: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                    then

                     A88: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

                    ( indx (D2,D1,j)) in ( Seg ( len ( lower_volume (f,D2)))) by A87, INTEGRA1:def 7;

                    then

                     A89: ( indx (D2,D1,j)) <= ( len ( lower_volume (f,D2))) by FINSEQ_1: 1;

                    then

                     A90: (( indx (D2,D1,j1)) + 1) <= ( len ( lower_volume (f,D2))) by A53, XXREAL_0: 2;

                    then (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( lower_volume (f,D2)))) by A82, FINSEQ_1: 1;

                    then

                     A91: (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 7;

                    then

                     A92: (( indx (D2,D1,j1)) + 1) in ( dom D2) by FINSEQ_1:def 3;

                    (( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) = (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) by A53, XREAL_1: 233;

                    then ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) <= 2 by A55;

                    then

                     A93: ( len ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) <= 2 by A53, A88, A89, A82, A90, FINSEQ_6: 118;

                    ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) >= ( 0 + 1) by XREAL_1: 6;

                    then

                     A94: 1 <= ( len ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) by A53, A88, A89, A82, A90, FINSEQ_6: 118;

                    now

                      per cases by A94, A93, Lm2;

                        suppose

                         A95: ( len ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 1;

                        ( upper_bound ( divset (D1,j))) = (D1 . j) by A42, A44, INTEGRA1:def 4;

                        then

                         A96: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A13, A42, INTEGRA1:def 19;

                        ( lower_bound ( divset (D1,j))) = (D1 . j1) by A42, A44, INTEGRA1:def 4;

                        then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A13, A45, INTEGRA1:def 19;

                        then

                         A97: ( divset (D1,j)) = [.(D2 . ( indx (D2,D1,j1))), (D2 . ( indx (D2,D1,j))).] by A96, INTEGRA1: 4;

                        

                         A98: ( delta D1) >= 0 by Th9;

                        

                         A99: (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                        

                         A100: ( indx (D2,D1,j)) in ( dom D2) by A13, A42, INTEGRA1:def 19;

                        ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 1 by A53, A88, A89, A82, A90, A95, FINSEQ_6: 118;

                        then

                         A101: (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) = 0 by A53, XREAL_1: 233;

                        then ( indx (D2,D1,j)) <> 1 by A49, FINSEQ_1: 1;

                        then

                         A102: ( upper_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . ( indx (D2,D1,j))) by A100, INTEGRA1:def 4;

                        (( indx (D2,D1,j)) - 1) = ( indx (D2,D1,j1)) by A101;

                        then ( lower_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . ( indx (D2,D1,j1))) by A50, A101, A100, INTEGRA1:def 4;

                        then

                         A103: ( divset (D2,( indx (D2,D1,j)))) = ( divset (D1,j)) by A97, A102, INTEGRA1: 4;

                        reconsider lv = (( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) as Element of REAL by XREAL_0:def 1;

                        (( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) by A88, A89, A82, A90, FINSEQ_6: 118;

                        then ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*lv*> by A95, FINSEQ_1: 40;

                        

                        then ( Sum ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = (( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) by FINSOP_1: 11

                        .= (( lower_bound ( rng (f | ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))) by A92, INTEGRA1:def 7

                        .= ( Sum ( mid (( lower_volume (f,D1)),j,j))) by A42, A86, A101, A103, INTEGRA1:def 7;

                        hence thesis by A98, A99;

                      end;

                        suppose

                         A104: ( len ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 2;

                        

                         A105: (( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) by A88, A89, A82, A90, FINSEQ_6: 118;

                        

                         A106: (2 + (( indx (D2,D1,j1)) + 1)) >= ( 0 + 1) by XREAL_1: 7;

                        (( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 2) = ( LVf(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) -' 1)) by A53, A88, A89, A82, A90, A104, FINSEQ_6: 118

                        .= ( LVf(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) - 1)) by A106, XREAL_1: 233

                        .= ( LVf(D2) . (( indx (D2,D1,j1)) + (1 + 1)));

                        then ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*(( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)), (( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 2))*> by A104, A105, FINSEQ_1: 44;

                        then

                         A107: ( Sum ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) + (( lower_volume (f,D2)) . (( indx (D2,D1,j1)) + 2))) by RVSUM_1: 77;

                        

                         A108: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) >= 0 by INTEGRA1: 9;

                        ( upper_bound ( divset (D1,j))) = (D1 . j) by A42, A44, INTEGRA1:def 4;

                        then

                         A109: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A13, A42, INTEGRA1:def 19;

                        

                         A110: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) >= 0 by INTEGRA1: 9;

                        ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A53, A88, A89, A82, A90, A104, FINSEQ_6: 118;

                        then

                         A111: ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A53, XREAL_1: 233;

                        then

                         A112: (( indx (D2,D1,j1)) + 2) in ( dom D2) by A13, A42, INTEGRA1:def 19;

                        ( lower_bound ( divset (D1,j))) = (D1 . j1) by A42, A44, INTEGRA1:def 4;

                        then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A13, A45, INTEGRA1:def 19;

                        then

                         A113: ( vol ( divset (D1,j))) = ((((D2 . (( indx (D2,D1,j1)) + 2)) - (D2 . (( indx (D2,D1,j1)) + 1))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A109, A111, INTEGRA1:def 5;

                        (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( lower_volume (f,D2)))) by A82, A90, FINSEQ_1: 1;

                        then (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 7;

                        then

                         A114: (( indx (D2,D1,j1)) + 1) in ( dom D2) by FINSEQ_1:def 3;

                        

                         A115: (( indx (D2,D1,j1)) + 1) <> 1 by A50, NAT_1: 13;

                        then

                         A116: ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A114, INTEGRA1:def 4;

                        ((( indx (D2,D1,j1)) + 1) - 1) = (( indx (D2,D1,j1)) + 0 );

                        then

                         A117: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . ( indx (D2,D1,j1))) by A114, A115, INTEGRA1:def 4;

                        

                         A118: ((( indx (D2,D1,j1)) + 1) + 1) > 1 by A82, NAT_1: 13;

                        ((( indx (D2,D1,j1)) + 2) - 1) = (( indx (D2,D1,j1)) + 1);

                        then

                         A119: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A112, A118, INTEGRA1:def 4;

                        ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 2)) by A112, A118, INTEGRA1:def 4;

                        

                        then ( vol ( divset (D1,j))) = ((( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A119, A113, INTEGRA1:def 5

                        .= (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) - ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A117, A116;

                        then

                         A120: ( vol ( divset (D1,j))) = (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) by INTEGRA1:def 5;

                        then

                         A121: (( lower_volume (f,D1)) . j) = (( lower_bound ( rng (f | ( divset (D1,j))))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) by A42, INTEGRA1:def 7;

                        

                         A122: (( Sum ( mid ( LVf(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) - ( Sum ( mid ( LVf(D1),j,j)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))))

                        proof

                          set ID2 = (( indx (D2,D1,j1)) + 2);

                          set ID1 = (( indx (D2,D1,j1)) + 1);

                          set B = ( vol ( divset (D2,ID1)));

                          set C = ( vol ( divset (D2,ID2)));

                          ( divset (D1,j)) c= A by A42, INTEGRA1: 8;

                          then

                           A123: ( lower_bound ( rng (f | ( divset (D1,j))))) >= ( lower_bound ( rng f)) by A1, Lm4;

                          ID1 in ( dom D2) by A91, FINSEQ_1:def 3;

                          then ( divset (D2,ID1)) c= A by INTEGRA1: 8;

                          then ( lower_bound ( rng (f | ( divset (D2,ID1))))) <= ( upper_bound ( rng f)) by A1, Lm4;

                          then

                           A124: (( lower_bound ( rng (f | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1)))) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1)))) by A108, XREAL_1: 64;

                          ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A53, A88, A89, A82, A90, A104, FINSEQ_6: 118;

                          then

                           A125: ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A53, XREAL_1: 233;

                          

                           A126: ( indx (D2,D1,j)) in ( dom D2) by A13, A42, INTEGRA1:def 19;

                          then ( divset (D2,ID2)) c= A by A125, INTEGRA1: 8;

                          then

                           A127: ( lower_bound ( rng (f | ( divset (D2,ID2))))) <= ( upper_bound ( rng f)) by A1, Lm4;

                          reconsider A = ( lower_bound ( rng (f | ( divset (D1,j))))) as Real;

                          

                           A128: ((( lower_volume (f,D1)) . j) - (A * B)) = (A * C) by A121;

                          (( lower_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID2)))) >= (( lower_bound ( rng f)) * ( vol ( divset (D2,ID2)))) by A110, A123, XREAL_1: 64;

                          then ( Sum ( mid ( LVf(D1),j,j))) >= ((( lower_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) + (( lower_bound ( rng f)) * ( vol ( divset (D2,ID2))))) by A86, A128, XREAL_1: 19;

                          then

                           A129: (( Sum ( mid ( LVf(D1),j,j))) - (( lower_bound ( rng f)) * ( vol ( divset (D2,ID2))))) >= (( lower_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) by XREAL_1: 19;

                          (( lower_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) >= (( lower_bound ( rng f)) * ( vol ( divset (D2,ID1)))) by A108, A123, XREAL_1: 64;

                          then (( Sum ( mid ( LVf(D1),j,j))) - (( lower_bound ( rng f)) * ( vol ( divset (D2,ID2))))) >= (( lower_bound ( rng f)) * ( vol ( divset (D2,ID1)))) by A129, XXREAL_0: 2;

                          then

                           A130: ( Sum ( mid ( LVf(D1),j,j))) >= ((( lower_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng f)) * ( vol ( divset (D2,ID1))))) by XREAL_1: 19;

                          ( Sum ( mid ( LVf(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( lower_bound ( rng (f | ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) + ( LVf(D2) . (( indx (D2,D1,j1)) + 1))) by A107, A126, A125, INTEGRA1:def 7

                          .= ((( lower_bound ( rng (f | ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) + (( lower_bound ( rng (f | ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A92, INTEGRA1:def 7;

                          then (( Sum ( mid ( LVf(D2),ID1,( indx (D2,D1,j))))) - (( lower_bound ( rng (f | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1))))) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,ID2)))) by A110, A127, XREAL_1: 64;

                          then ( Sum ( mid ( LVf(D2),ID1,( indx (D2,D1,j))))) <= ((( upper_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng (f | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1))))) by XREAL_1: 20;

                          then (( Sum ( mid ( LVf(D2),ID1,( indx (D2,D1,j))))) - (( upper_bound ( rng f)) * ( vol ( divset (D2,ID2))))) <= (( lower_bound ( rng (f | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1)))) by XREAL_1: 20;

                          then (( Sum ( mid ( LVf(D2),ID1,( indx (D2,D1,j))))) - (( upper_bound ( rng f)) * ( vol ( divset (D2,ID2))))) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1)))) by A124, XXREAL_0: 2;

                          then ( Sum ( mid ( LVf(D2),ID1,( indx (D2,D1,j))))) <= ((( upper_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1))))) by XREAL_1: 20;

                          then (( Sum ( mid ( LVf(D2),ID1,( indx (D2,D1,j))))) - ( Sum ( mid ( LVf(D1),j,j)))) <= (((( upper_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1))))) - ((( lower_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng f)) * ( vol ( divset (D2,ID1)))))) by A130, XREAL_1: 13;

                          hence thesis;

                        end;

                        (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                        then ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( vol ( divset (D1,j)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1)) by A42, Lm5, XREAL_1: 64;

                        hence thesis by A120, A122, XXREAL_0: 2;

                      end;

                    end;

                    hence thesis;

                  end;

                  j < (j + 1) by NAT_1: 13;

                  then

                   A131: j1 < j by XREAL_1: 19;

                  ( indx (D2,D1,j)) in ( dom D2) by A13, A42, INTEGRA1:def 19;

                  then

                   A132: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                  then

                   A133: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

                  

                   A134: ( indx (D2,D1,j1)) <= ( len D2) by A49, FINSEQ_1: 1;

                  then

                   A135: ( len (D2 | ( indx (D2,D1,j1)))) = ( indx (D2,D1,j1)) by FINSEQ_1: 59;

                  

                   A136: j1 in ( Seg ( len D1)) by A47, FINSEQ_1:def 3;

                  then

                   A137: j1 <= ( len D1) by FINSEQ_1: 1;

                  for x1 be object st x1 in ( rng (D1 | j1)) holds x1 in ( rng (D2 | ( indx (D2,D1,j1))))

                  proof

                    let x1 be object;

                    assume x1 in ( rng (D1 | j1));

                    then

                    consider k such that

                     A138: k in ( dom (D1 | j1)) and

                     A139: x1 = ((D1 | j1) . k) by PARTFUN1: 3;

                    k in ( Seg ( len (D1 | j1))) by A138, FINSEQ_1:def 3;

                    then

                     A140: k in ( Seg j1) by A137, FINSEQ_1: 59;

                    then

                     A141: k in ( dom D1) by A45, RFINSEQ: 6;

                    k <= j1 by A140, FINSEQ_1: 1;

                    then (D1 . k) <= (D1 . j1) by A47, A141, SEQ_4: 137;

                    then (D2 . ( indx (D2,D1,k))) <= (D1 . j1) by A13, A141, INTEGRA1:def 19;

                    then

                     A142: (D2 . ( indx (D2,D1,k))) <= (D2 . ( indx (D2,D1,j1))) by A13, A47, INTEGRA1:def 19;

                    

                     A143: ((D1 | j1) . k) = (D1 . k) by A45, A140, RFINSEQ: 6;

                    (D1 . k) in ( rng D1) by A141, FUNCT_1:def 3;

                    then x1 in ( rng D2) by A14, A139, A143, XBOOLE_0:def 3;

                    then

                    consider n such that

                     A144: n in ( dom D2) and

                     A145: x1 = (D2 . n) by PARTFUN1: 3;

                    (D2 . ( indx (D2,D1,k))) = (D2 . n) by A13, A139, A143, A141, A145, INTEGRA1:def 19;

                    then

                     A146: n <= ( indx (D2,D1,j1)) by A48, A144, A142, SEQM_3:def 1;

                    1 <= n by A144, FINSEQ_3: 25;

                    then

                     A147: n in ( Seg ( indx (D2,D1,j1))) by A146, FINSEQ_1: 1;

                    then n in ( Seg ( len (D2 | ( indx (D2,D1,j1))))) by A134, FINSEQ_1: 59;

                    then

                     A148: n in ( dom (D2 | ( indx (D2,D1,j1)))) by FINSEQ_1:def 3;

                    (D2 . n) = ((D2 | ( indx (D2,D1,j1))) . n) by A48, A147, RFINSEQ: 6;

                    hence thesis by A145, A148, FUNCT_1:def 3;

                  end;

                  then

                   A149: ( rng (D1 | j1)) c= ( rng (D2 | ( indx (D2,D1,j1))));

                  

                   A150: 1 <= j1 by A136, FINSEQ_1: 1;

                  ( lower_bound ( divset (D1,j))) <= (D . 1) by A43, INTEGRA2: 1;

                  then

                   A151: (D1 . j1) <= (D . 1) by A42, A44, INTEGRA1:def 4;

                  for x1 be object st x1 in ( rng (D2 | ( indx (D2,D1,j1)))) holds x1 in ( rng (D1 | j1))

                  proof

                    let x1 be object;

                    assume x1 in ( rng (D2 | ( indx (D2,D1,j1))));

                    then

                    consider k such that

                     A152: k in ( dom (D2 | ( indx (D2,D1,j1)))) and

                     A153: x1 = ((D2 | ( indx (D2,D1,j1))) . k) by PARTFUN1: 3;

                    k in ( Seg ( len (D2 | ( indx (D2,D1,j1))))) by A152, FINSEQ_1:def 3;

                    then

                     A154: k in ( Seg ( indx (D2,D1,j1))) by A134, FINSEQ_1: 59;

                    then

                     A155: k in ( dom D2) by A48, RFINSEQ: 6;

                    

                     A156: ( len (D1 | j1)) = j1 by A137, FINSEQ_1: 59;

                    k <= ( indx (D2,D1,j1)) by A154, FINSEQ_1: 1;

                    then (D2 . k) <= (D2 . ( indx (D2,D1,j1))) by A48, A155, SEQ_4: 137;

                    then

                     A157: (D2 . k) <= (D1 . j1) by A13, A47, INTEGRA1:def 19;

                    

                     A158: (D2 . k) in ( rng D1) implies (D2 . k) in ( rng (D1 | j1))

                    proof

                      assume (D2 . k) in ( rng D1);

                      then

                      consider m such that

                       A159: m in ( dom D1) and

                       A160: (D2 . k) = (D1 . m) by PARTFUN1: 3;

                      m in ( Seg ( len D1)) by A159, FINSEQ_1:def 3;

                      then

                       A161: 1 <= m by FINSEQ_1: 1;

                      

                       A162: m <= j1 by A45, A157, A159, A160, SEQM_3:def 1;

                      then m in ( Seg j1) by A161, FINSEQ_1: 1;

                      then

                       A163: (D2 . k) = ((D1 | j1) . m) by A45, A160, RFINSEQ: 6;

                      m in ( dom (D1 | j1)) by A156, A161, A162, FINSEQ_3: 25;

                      hence thesis by A163, FUNCT_1:def 3;

                    end;

                    

                     A164: (D2 . k) in ( rng D) implies (D2 . k) = (D1 . j1)

                    proof

                      assume (D2 . k) in ( rng D);

                      then

                      consider n such that

                       A165: n in ( dom D) and

                       A166: (D2 . k) = (D . n) by PARTFUN1: 3;

                      1 <= n by A165, FINSEQ_3: 25;

                      then (D . 1) <= (D2 . k) by A41, A165, A166, SEQ_4: 137;

                      then (D1 . j1) <= (D2 . k) by A151, XXREAL_0: 2;

                      hence thesis by A157, XXREAL_0: 1;

                    end;

                    

                     A167: (D2 . k) in ( rng D) implies (D2 . k) in ( rng (D1 | j1))

                    proof

                      j1 in ( Seg ( len (D1 | j1))) by A150, A156, FINSEQ_1: 1;

                      then j1 in ( dom (D1 | j1)) by FINSEQ_1:def 3;

                      then

                       A168: ((D1 | j1) . j1) in ( rng (D1 | j1)) by FUNCT_1:def 3;

                      assume

                       A169: (D2 . k) in ( rng D);

                      j1 in ( Seg j1) by A150, FINSEQ_1: 1;

                      hence thesis by A45, A164, A169, A168, RFINSEQ: 6;

                    end;

                    (D2 . k) in ( rng D2) by A155, FUNCT_1:def 3;

                    hence thesis by A14, A48, A153, A154, A167, A158, RFINSEQ: 6, XBOOLE_0:def 3;

                  end;

                  then ( rng (D2 | ( indx (D2,D1,j1)))) c= ( rng (D1 | j1));

                  then

                   A170: ( rng (D2 | ( indx (D2,D1,j1)))) = ( rng (D1 | j1)) by A149, XBOOLE_0:def 10;

                  ( mid (D1,1,j1)) is increasing by A42, A44, A150, INTEGRA1: 7, INTEGRA1: 35;

                  then

                   A171: (D1 | j1) is increasing by A150, FINSEQ_6: 116;

                  then

                   A172: (D2 | ( indx (D2,D1,j1))) = (D1 | j1) by A51, A170, Th6;

                  

                   A173: for k st 1 <= k & k <= j1 holds k = ( indx (D2,D1,k))

                  proof

                    let k;

                    assume that

                     A174: 1 <= k and

                     A175: k <= j1;

                    assume

                     A176: k <> ( indx (D2,D1,k));

                    now

                      per cases by A176, XXREAL_0: 1;

                        suppose

                         A177: k > ( indx (D2,D1,k));

                        k <= ( len D1) by A137, A175, XXREAL_0: 2;

                        then

                         A178: k in ( dom D1) by A174, FINSEQ_3: 25;

                        then ( indx (D2,D1,k)) in ( dom D2) by A13, INTEGRA1:def 19;

                        then ( indx (D2,D1,k)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                        then

                         A179: 1 <= ( indx (D2,D1,k)) by FINSEQ_1: 1;

                        

                         A180: ( indx (D2,D1,k)) < j1 by A175, A177, XXREAL_0: 2;

                        then

                         A181: ( indx (D2,D1,k)) in ( Seg j1) by A179, FINSEQ_1: 1;

                        ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A13, A45, A175, A178, Th7;

                        then ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A179, FINSEQ_1: 1;

                        then

                         A182: ((D2 | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) = (D2 . ( indx (D2,D1,k))) by A48, RFINSEQ: 6;

                        ( indx (D2,D1,k)) <= ( len D1) by A137, A180, XXREAL_0: 2;

                        then ( indx (D2,D1,k)) in ( dom D1) by A179, FINSEQ_3: 25;

                        then

                         A183: (D1 . k) > (D1 . ( indx (D2,D1,k))) by A177, A178, SEQM_3:def 1;

                        (D1 . k) = (D2 . ( indx (D2,D1,k))) by A13, A178, INTEGRA1:def 19;

                        hence contradiction by A45, A172, A182, A183, A181, RFINSEQ: 6;

                      end;

                        suppose

                         A184: k < ( indx (D2,D1,k));

                        k <= ( len D1) by A137, A175, XXREAL_0: 2;

                        then

                         A185: k in ( dom D1) by A174, FINSEQ_3: 25;

                        then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A13, A45, A175, Th7;

                        then

                         A186: k <= ( indx (D2,D1,j1)) by A184, XXREAL_0: 2;

                        then k <= ( len D2) by A134, XXREAL_0: 2;

                        then

                         A187: k in ( dom D2) by A174, FINSEQ_3: 25;

                        k in ( Seg j1) by A174, A175, FINSEQ_1: 1;

                        then

                         A188: (D1 . k) = ((D1 | j1) . k) by A47, RFINSEQ: 6;

                        ( indx (D2,D1,k)) in ( dom D2) by A13, A185, INTEGRA1:def 19;

                        then

                         A189: (D2 . k) < (D2 . ( indx (D2,D1,k))) by A184, A187, SEQM_3:def 1;

                        

                         A190: k in ( Seg ( indx (D2,D1,j1))) by A174, A186, FINSEQ_1: 1;

                        (D1 . k) = (D2 . ( indx (D2,D1,k))) by A13, A185, INTEGRA1:def 19;

                        hence contradiction by A48, A172, A188, A189, A190, RFINSEQ: 6;

                      end;

                    end;

                    hence contradiction;

                  end;

                  

                   A191: for k be Nat st 1 <= k & k <= ( len (( lower_volume (f,D1)) | j1)) holds ((( lower_volume (f,D1)) | j1) . k) = ((( lower_volume (f,D2)) | ( indx (D2,D1,j1))) . k)

                  proof

                    ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A48, FINSEQ_1:def 3;

                    then ( indx (D2,D1,j1)) in ( Seg ( len ( lower_volume (f,D2)))) by INTEGRA1:def 7;

                    then

                     A192: ( indx (D2,D1,j1)) in ( dom ( lower_volume (f,D2))) by FINSEQ_1:def 3;

                    let k be Nat;

                    assume that

                     A193: 1 <= k and

                     A194: k <= ( len (( lower_volume (f,D1)) | j1));

                    reconsider k as Element of NAT by ORDINAL1:def 12;

                    

                     A195: ( len ( lower_volume (f,D1))) = ( len D1) by INTEGRA1:def 7;

                    then

                     A196: k <= j1 by A137, A194, FINSEQ_1: 59;

                    then k <= ( len D1) by A137, XXREAL_0: 2;

                    then

                     A197: k in ( Seg ( len D1)) by A193, FINSEQ_1: 1;

                    then

                     A198: k in ( dom D1) by FINSEQ_1:def 3;

                    then

                     A199: ( indx (D2,D1,k)) in ( dom D2) by A13, INTEGRA1:def 19;

                    

                     A200: k in ( Seg j1) by A193, A196, FINSEQ_1: 1;

                    then ( indx (D2,D1,k)) in ( Seg j1) by A173, A193, A196;

                    then

                     A201: ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A150, A173;

                    then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by FINSEQ_1: 1;

                    then

                     A202: ( indx (D2,D1,k)) <= ( len D2) by A134, XXREAL_0: 2;

                    

                     A203: (D1 . k) = (D2 . ( indx (D2,D1,k))) by A13, A198, INTEGRA1:def 19;

                    

                     A204: ( lower_bound ( divset (D1,k))) = ( lower_bound ( divset (D2,( indx (D2,D1,k))))) & ( upper_bound ( divset (D1,k))) = ( upper_bound ( divset (D2,( indx (D2,D1,k)))))

                    proof

                      per cases ;

                        suppose

                         A205: k = 1;

                        then

                         A206: ( upper_bound ( divset (D1,k))) = (D1 . k) by A198, INTEGRA1:def 4;

                        

                         A207: ( lower_bound ( divset (D1,k))) = ( lower_bound A) by A198, A205, INTEGRA1:def 4;

                        ( indx (D2,D1,k)) = 1 by A150, A173, A205;

                        hence thesis by A199, A203, A207, A206, INTEGRA1:def 4;

                      end;

                        suppose

                         A208: k <> 1;

                        then

                        reconsider k1 = (k - 1) as Element of NAT by A198, INTEGRA1: 7;

                        k <= (k + 1) by NAT_1: 11;

                        then k1 <= k by XREAL_1: 20;

                        then

                         A209: k1 <= j1 by A196, XXREAL_0: 2;

                        

                         A210: (k - 1) in ( dom D1) by A198, A208, INTEGRA1: 7;

                        then k1 in ( Seg ( len D1)) by FINSEQ_1:def 3;

                        then 1 <= k1 by FINSEQ_1: 1;

                        then k1 = ( indx (D2,D1,k1)) by A173, A209;

                        then

                         A211: (D2 . (( indx (D2,D1,k)) - 1)) = (D2 . ( indx (D2,D1,k1))) by A173, A193, A196;

                        

                         A212: ( indx (D2,D1,k)) <> 1 by A173, A193, A196, A208;

                        then

                         A213: ( lower_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . (( indx (D2,D1,k)) - 1)) by A199, INTEGRA1:def 4;

                        

                         A214: ( upper_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . ( indx (D2,D1,k))) by A199, A212, INTEGRA1:def 4;

                        

                         A215: ( upper_bound ( divset (D1,k))) = (D1 . k) by A198, A208, INTEGRA1:def 4;

                        ( lower_bound ( divset (D1,k))) = (D1 . (k - 1)) by A198, A208, INTEGRA1:def 4;

                        hence thesis by A13, A198, A215, A210, A213, A214, A211, INTEGRA1:def 19;

                      end;

                    end;

                    ( divset (D2,( indx (D2,D1,k)))) = [.( lower_bound ( divset (D2,( indx (D2,D1,k))))), ( upper_bound ( divset (D2,( indx (D2,D1,k))))).] by INTEGRA1: 4;

                    then

                     A216: ( divset (D1,k)) = ( divset (D2,( indx (D2,D1,k)))) by A204, INTEGRA1: 4;

                    

                     A217: k in ( dom D1) by A197, FINSEQ_1:def 3;

                    j1 in ( Seg ( len ( lower_volume (f,D1)))) by A45, A195, FINSEQ_1:def 3;

                    then j1 in ( dom ( lower_volume (f,D1))) by FINSEQ_1:def 3;

                    

                    then

                     A218: ((( lower_volume (f,D1)) | j1) . k) = (( lower_volume (f,D1)) . k) by A200, RFINSEQ: 6

                    .= (( lower_bound ( rng (f | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A217, A216, INTEGRA1:def 7;

                    1 <= ( indx (D2,D1,k)) by A173, A193, A196;

                    then ( indx (D2,D1,k)) in ( Seg ( len D2)) by A202, FINSEQ_1: 1;

                    then

                     A219: ( indx (D2,D1,k)) in ( dom D2) by FINSEQ_1:def 3;

                    ((( lower_volume (f,D2)) | ( indx (D2,D1,j1))) . k) = ((( lower_volume (f,D2)) | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) by A173, A193, A196

                    .= (( lower_volume (f,D2)) . ( indx (D2,D1,k))) by A201, A192, RFINSEQ: 6

                    .= (( lower_bound ( rng (f | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A219, INTEGRA1:def 7;

                    hence thesis by A218;

                  end;

                  ( indx (D2,D1,j1)) in ( dom D2) by A13, A47, INTEGRA1:def 19;

                  then ( indx (D2,D1,j1)) <= ( len D2) by FINSEQ_3: 25;

                  then

                   A220: ( indx (D2,D1,j1)) <= ( len ( lower_volume (f,D2))) by INTEGRA1:def 7;

                  j1 <= ( len D1) by A47, FINSEQ_3: 25;

                  then

                   A221: j1 <= ( len ( lower_volume (f,D1))) by INTEGRA1:def 7;

                  ( len (D2 | ( indx (D2,D1,j1)))) = ( len (D1 | j1)) by A51, A171, A170, Th6;

                  then ( indx (D2,D1,j1)) = j1 by A137, A135, FINSEQ_1: 59;

                  then ( len (( lower_volume (f,D1)) | j1)) = ( indx (D2,D1,j1)) by A221, FINSEQ_1: 59;

                  then ( len (( lower_volume (f,D1)) | j1)) = ( len (( lower_volume (f,D2)) | ( indx (D2,D1,j1)))) by A220, FINSEQ_1: 59;

                  then

                   A222: (( lower_volume (f,D2)) | ( indx (D2,D1,j1))) = (( lower_volume (f,D1)) | j1) by A191, FINSEQ_1: 14;

                  

                   A223: j in ( Seg ( len D1)) by A42, FINSEQ_1:def 3;

                  then

                   A224: 1 <= j by FINSEQ_1: 1;

                  ( indx (D2,D1,j)) in ( Seg ( len LVf(D2))) by A132, INTEGRA1:def 7;

                  then

                   A225: ( indx (D2,D1,j)) in ( dom LVf(D2)) by FINSEQ_1:def 3;

                  ( indx (D2,D1,j)) <= ( len D2) by A132, FINSEQ_1: 1;

                  then

                   A226: ( indx (D2,D1,j)) <= ( len LVf(D2)) by INTEGRA1:def 7;

                  j in ( Seg ( len LVf(D1))) by A223, INTEGRA1:def 7;

                  then

                   A227: j in ( dom LVf(D1)) by FINSEQ_1:def 3;

                  j <= ( len D1) by A223, FINSEQ_1: 1;

                  then

                   A228: j <= ( len LVf(D1)) by INTEGRA1:def 7;

                  j1 in ( Seg ( len D1)) by A45, FINSEQ_1:def 3;

                  then j1 in ( Seg ( len LVf(D1))) by INTEGRA1:def 7;

                  then j1 in ( dom LVf(D1)) by FINSEQ_1:def 3;

                  then PLf(D1,j1) = ( Sum ( LVf(D1) | j1)) by INTEGRA1:def 20;

                  

                  then ( PLf(D1,j1) + ( Sum ( mid ( LVf(D1),j,j)))) = ( Sum (( LVf(D1) | j1) ^ ( mid ( LVf(D1),j,j)))) by RVSUM_1: 75

                  .= ( Sum (( mid ( LVf(D1),1,j1)) ^ ( mid ( LVf(D1),(j1 + 1),j)))) by A150, FINSEQ_6: 116

                  .= ( Sum ( mid ( LVf(D1),1,j))) by A150, A228, A131, INTEGRA2: 4

                  .= ( Sum ( LVf(D1) | j)) by A224, FINSEQ_6: 116;

                  then

                   A229: ( PLf(D1,j1) + ( Sum ( mid (( lower_volume (f,D1)),j,j)))) = PLf(D1,j) by A227, INTEGRA1:def 20;

                  ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A48, FINSEQ_1:def 3;

                  then ( indx (D2,D1,j1)) in ( Seg ( len LVf(D2))) by INTEGRA1:def 7;

                  then ( indx (D2,D1,j1)) in ( dom LVf(D2)) by FINSEQ_1:def 3;

                  then PLf(D2,indx) = ( Sum ( LVf(D2) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20;

                  

                  then ( PLf(D2,indx) + ( Sum ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = ( Sum (( LVf(D2) | ( indx (D2,D1,j1))) ^ ( mid ( LVf(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by RVSUM_1: 75

                  .= ( Sum (( mid ( LVf(D2),1,( indx (D2,D1,j1)))) ^ ( mid ( LVf(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by A50, FINSEQ_6: 116

                  .= ( Sum ( mid ( LVf(D2),1,( indx (D2,D1,j))))) by A50, A52, A226, INTEGRA2: 4

                  .= ( Sum ( LVf(D2) | ( indx (D2,D1,j)))) by A133, FINSEQ_6: 116;

                  then

                   A230: ( PLf(D2,indx) + ( Sum ( mid (( lower_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = PLf(D2,indx) by A225, INTEGRA1:def 20;

                  ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A48, FINSEQ_1:def 3;

                  then ( indx (D2,D1,j1)) in ( Seg ( len ( lower_volume (f,D2)))) by INTEGRA1:def 7;

                  then ( indx (D2,D1,j1)) in ( dom ( lower_volume (f,D2))) by FINSEQ_1:def 3;

                  

                  then PLf(D2,indx) = ( Sum (( lower_volume (f,D2)) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20

                  .= PLf(D1,j1) by A222, A46, INTEGRA1:def 20;

                  hence thesis by A54, A230, A229;

                end;

                hence thesis by A42, A43;

              end;

              reconsider i as non zero Element of NAT by A18, FINSEQ_1: 1;

              

               A231: for i be non zero Nat st P[i] holds P[(i + 1)]

              proof

                let i be non zero Nat;

                

                 A232: i >= 1 by NAT_1: 14;

                assume

                 A233: P[i];

                 P[(i + 1)]

                proof

                  

                   A234: i <= (i + 1) by NAT_1: 11;

                  assume

                   A235: (i + 1) in ( dom D);

                  then

                  consider j such that

                   A236: j in ( dom D1) and

                   A237: (D . (i + 1)) in ( divset (D1,j)) by Th3, INTEGRA1: 6;

                  

                   A238: (D2 . ( indx (D2,D1,j))) = (D1 . j) by A13, A236, INTEGRA1:def 19;

                  (i + 1) <= ( len D) by A235, FINSEQ_3: 25;

                  then i <= ( len D) by A234, XXREAL_0: 2;

                  then

                   A239: i in ( Seg ( len D)) by A232, FINSEQ_1: 1;

                  then

                   A240: i in ( dom D) by FINSEQ_1:def 3;

                  consider n1 be Element of NAT such that

                   A241: n1 in ( dom D1) and

                   A242: (D . i) in ( divset (D1,n1)) and

                   A243: ( PLf(D2,indx) - PLf(D1,n1)) <= ((i * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A233, A239, FINSEQ_1:def 3;

                  

                   A244: 1 <= (n1 + 1) by NAT_1: 12;

                  

                   A245: n1 < j

                  proof

                    assume

                     A246: n1 >= j;

                    now

                      per cases by A246, XXREAL_0: 1;

                        suppose

                         A247: n1 = j;

                        (D . i) in ( rng D) by A240, FUNCT_1:def 3;

                        then

                         A248: (D . i) in (( rng D) /\ ( divset (D1,j))) by A242, A247, XBOOLE_0:def 4;

                        (D . (i + 1)) in ( rng D) by A235, FUNCT_1:def 3;

                        then

                         A249: (D . (i + 1)) in (( rng D) /\ ( divset (D1,j))) by A237, XBOOLE_0:def 4;

                        (i + 1) > i by XREAL_1: 29;

                        hence contradiction by A11, A235, A236, A240, A248, A249, Th5, SEQ_4: 138;

                      end;

                        suppose n1 > j;

                        then

                         A250: n1 >= (j + 1) by NAT_1: 13;

                        then

                         A251: (n1 - 1) >= j by XREAL_1: 19;

                        1 <= j by A236, FINSEQ_3: 25;

                        then (1 + 1) <= (j + 1) by XREAL_1: 6;

                        then

                         A252: n1 <> 1 by A250, XXREAL_0: 2;

                        ( lower_bound ( divset (D1,n1))) <= (D . i) by A242, INTEGRA2: 1;

                        then

                         A253: (D . i) >= (D1 . (n1 - 1)) by A241, A252, INTEGRA1:def 4;

                        (n1 - 1) in ( dom D1) by A241, A252, INTEGRA1: 7;

                        then (D1 . j) <= (D1 . (n1 - 1)) by A236, A251, SEQ_4: 137;

                        then

                         A254: (D . i) >= (D1 . j) by A253, XXREAL_0: 2;

                        

                         A255: i < (i + 1) by XREAL_1: 29;

                        

                         A256: ( upper_bound ( divset (D1,j))) = (D1 . j)

                        proof

                          per cases ;

                            suppose j = 1;

                            hence thesis by A236, INTEGRA1:def 4;

                          end;

                            suppose j <> 1;

                            hence thesis by A236, INTEGRA1:def 4;

                          end;

                        end;

                        (D . (i + 1)) <= ( upper_bound ( divset (D1,j))) by A237, INTEGRA2: 1;

                        then (D . i) >= (D . (i + 1)) by A256, A254, XXREAL_0: 2;

                        hence contradiction by A235, A240, A255, SEQM_3:def 1;

                      end;

                    end;

                    hence thesis;

                  end;

                  then

                   A257: (n1 + 1) <= j by NAT_1: 13;

                  

                   A258: 1 <= n1 by A241, FINSEQ_3: 25;

                  

                   A259: ( indx (D2,D1,n1)) in ( dom D2) by A13, A241, INTEGRA1:def 19;

                  then

                   A260: 1 <= ( indx (D2,D1,n1)) by FINSEQ_3: 25;

                  

                   A261: ( indx (D2,D1,j)) in ( dom D2) by A13, A236, INTEGRA1:def 19;

                  then

                   A262: 1 <= ( indx (D2,D1,j)) by FINSEQ_3: 25;

                  

                   A263: ( indx (D2,D1,j)) <= ( len D2) by A261, FINSEQ_3: 25;

                  then

                   A264: ( indx (D2,D1,j)) <= ( len LVf(D2)) by INTEGRA1:def 7;

                  

                   A265: 1 <= j by A236, FINSEQ_3: 25;

                  

                   A266: j <= ( len D1) by A236, FINSEQ_3: 25;

                  then

                   A267: (n1 + 1) <= ( len D1) by A257, XXREAL_0: 2;

                  then

                   A268: (n1 + 1) in ( dom D1) by A244, FINSEQ_3: 25;

                  then

                   A269: ( indx (D2,D1,(n1 + 1))) in ( dom D2) by A13, INTEGRA1:def 19;

                  then

                   A270: 1 <= ( indx (D2,D1,(n1 + 1))) by FINSEQ_3: 25;

                  

                   A271: (D2 . ( indx (D2,D1,(n1 + 1)))) = (D1 . (n1 + 1)) by A13, A268, INTEGRA1:def 19;

                  then (D2 . ( indx (D2,D1,(n1 + 1)))) <= (D2 . ( indx (D2,D1,j))) by A236, A257, A268, A238, SEQ_4: 137;

                  then

                   A272: ( indx (D2,D1,(n1 + 1))) <= ( indx (D2,D1,j)) by A269, A261, SEQM_3:def 1;

                  then (1 + ( indx (D2,D1,(n1 + 1)))) <= (( indx (D2,D1,j)) + 1) by XREAL_1: 6;

                  then 1 <= ((( indx (D2,D1,j)) + 1) - ( indx (D2,D1,(n1 + 1)))) by XREAL_1: 19;

                  

                  then

                   A273: (( mid (D2,( indx (D2,D1,(n1 + 1))),( indx (D2,D1,j)))) . 1) = (D2 . ((1 - 1) + ( indx (D2,D1,(n1 + 1))))) by A272, A270, A263, FINSEQ_6: 122

                  .= (D1 . (n1 + 1)) by A13, A268, INTEGRA1:def 19;

                  

                   A274: (D2 . ( indx (D2,D1,n1))) = (D1 . n1) by A13, A241, INTEGRA1:def 19;

                  

                   A275: j <= ( len LVf(D1)) by A266, INTEGRA1:def 7;

                  then j in ( Seg ( len LVf(D1))) by A265, FINSEQ_1: 1;

                  then

                   A276: j in ( dom LVf(D1)) by FINSEQ_1:def 3;

                  

                   A277: ( indx (D2,D1,(n1 + 1))) <= ( len D2) by A269, FINSEQ_3: 25;

                  n1 in ( Seg ( len D1)) by A241, FINSEQ_1:def 3;

                  then n1 in ( Seg ( len LVf(D1))) by INTEGRA1:def 7;

                  then n1 in ( dom LVf(D1)) by FINSEQ_1:def 3;

                  

                  then PLf(D1,n1) = ( Sum ( LVf(D1) | n1)) by INTEGRA1:def 20

                  .= ( Sum ( mid ( LVf(D1),1,n1))) by A258, FINSEQ_6: 116;

                  

                  then ( PLf(D1,n1) + ( Sum ( mid ( LVf(D1),(n1 + 1),j)))) = ( Sum (( mid ( LVf(D1),1,n1)) ^ ( mid ( LVf(D1),(n1 + 1),j)))) by RVSUM_1: 75

                  .= ( Sum ( mid ( LVf(D1),1,j))) by A245, A258, A275, INTEGRA2: 4

                  .= ( Sum ( LVf(D1) | j)) by A265, FINSEQ_6: 116;

                  then

                   A278: PLf(D1,j) = ( PLf(D1,n1) + ( Sum ( mid ( LVf(D1),(n1 + 1),j)))) by A276, INTEGRA1:def 20;

                  ( indx (D2,D1,j)) in ( Seg ( len D2)) by A261, FINSEQ_1:def 3;

                  then ( indx (D2,D1,j)) in ( Seg ( len LVf(D2))) by INTEGRA1:def 7;

                  then

                   A279: ( indx (D2,D1,j)) in ( dom LVf(D2)) by FINSEQ_1:def 3;

                  

                   A280: n1 >= 1 by A241, FINSEQ_3: 25;

                  

                   A281: (j - n1) >= 1 by A257, XREAL_1: 19;

                  (( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) - ( Sum ( mid ( LVf(D1),(n1 + 1),j)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))

                  proof

                    now

                      per cases by A257, XXREAL_0: 1;

                        suppose

                         A282: (n1 + 1) = j;

                        

                         A283: (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) <= 2

                        proof

                          

                           A284: ( upper_bound ( divset (D1,j))) = (D1 . j) by A236, A245, A280, INTEGRA1:def 4;

                          

                           A285: ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A236, A245, A280, INTEGRA1:def 4;

                          

                           A286: 1 <= (( indx (D2,D1,n1)) + 1) by A260, NAT_1: 13;

                          assume (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) > 2;

                          then

                           A287: (( indx (D2,D1,n1)) + 2) < ( indx (D2,D1,j)) by XREAL_1: 20;

                          then

                           A288: (( indx (D2,D1,n1)) + 2) <= ( len D2) by A263, XXREAL_0: 2;

                          

                           A289: (( indx (D2,D1,n1)) + 1) < (( indx (D2,D1,n1)) + 2) by XREAL_1: 6;

                          then

                           A290: ( indx (D2,D1,n1)) < (( indx (D2,D1,n1)) + 2) by NAT_1: 13;

                          then 1 <= (( indx (D2,D1,n1)) + 2) by A260, XXREAL_0: 2;

                          then

                           A291: (( indx (D2,D1,n1)) + 2) in ( dom D2) by A288, FINSEQ_3: 25;

                          then

                           A292: (D2 . ( indx (D2,D1,j))) >= (D2 . (( indx (D2,D1,n1)) + 2)) by A261, A287, SEQ_4: 137;

                          

                           A293: not (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D1)

                          proof

                            assume (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D1);

                            then

                            consider k1 be Element of NAT such that

                             A294: k1 in ( dom D1) and

                             A295: (D2 . (( indx (D2,D1,n1)) + 2)) = (D1 . k1) by PARTFUN1: 3;

                            (D2 . (( indx (D2,D1,n1)) + 2)) < (D2 . ( indx (D2,D1,j))) by A261, A287, A291, SEQM_3:def 1;

                            then

                             A296: k1 < j by A236, A238, A294, A295, SEQ_4: 137;

                            (D2 . ( indx (D2,D1,n1))) < (D2 . (( indx (D2,D1,n1)) + 2)) by A259, A290, A291, SEQM_3:def 1;

                            then n1 < k1 by A241, A274, A294, A295, SEQ_4: 137;

                            hence contradiction by A282, A296, NAT_1: 13;

                          end;

                          (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D2) by A291, FUNCT_1:def 3;

                          then

                           A297: (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D) by A14, A293, XBOOLE_0:def 3;

                          

                           A298: ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A236, A245, A280, INTEGRA1:def 4;

                          

                           A299: ( upper_bound ( divset (D1,j))) = (D1 . j) by A236, A245, A280, INTEGRA1:def 4;

                          (D2 . (( indx (D2,D1,n1)) + 2)) >= (D2 . ( indx (D2,D1,n1))) by A259, A290, A291, SEQ_4: 137;

                          then (D2 . (( indx (D2,D1,n1)) + 2)) in ( divset (D1,j)) by A274, A238, A282, A298, A284, A292, INTEGRA2: 1;

                          then

                           A300: (D2 . (( indx (D2,D1,n1)) + 2)) in (( rng D) /\ ( divset (D1,j))) by A297, XBOOLE_0:def 4;

                          

                           A301: (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j)) by A287, A289, XXREAL_0: 2;

                          then (( indx (D2,D1,n1)) + 1) <= ( len D2) by A263, XXREAL_0: 2;

                          then

                           A302: (( indx (D2,D1,n1)) + 1) in ( dom D2) by A286, FINSEQ_3: 25;

                          then

                           A303: (D2 . ( indx (D2,D1,j))) >= (D2 . (( indx (D2,D1,n1)) + 1)) by A261, A301, SEQ_4: 137;

                          

                           A304: ( indx (D2,D1,n1)) < (( indx (D2,D1,n1)) + 1) by NAT_1: 13;

                          

                           A305: not (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D1)

                          proof

                            assume (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D1);

                            then

                            consider k1 be Element of NAT such that

                             A306: k1 in ( dom D1) and

                             A307: (D2 . (( indx (D2,D1,n1)) + 1)) = (D1 . k1) by PARTFUN1: 3;

                            (D2 . (( indx (D2,D1,n1)) + 1)) < (D2 . ( indx (D2,D1,j))) by A261, A301, A302, SEQM_3:def 1;

                            then

                             A308: k1 < j by A236, A238, A306, A307, SEQ_4: 137;

                            (D2 . ( indx (D2,D1,n1))) < (D2 . (( indx (D2,D1,n1)) + 1)) by A259, A304, A302, SEQM_3:def 1;

                            then n1 < k1 by A241, A274, A306, A307, SEQ_4: 137;

                            hence contradiction by A282, A308, NAT_1: 13;

                          end;

                          (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D2) by A302, FUNCT_1:def 3;

                          then

                           A309: (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D) by A14, A305, XBOOLE_0:def 3;

                          (D2 . (( indx (D2,D1,n1)) + 1)) >= (D2 . ( indx (D2,D1,n1))) by A259, A304, A302, SEQ_4: 137;

                          then (D2 . (( indx (D2,D1,n1)) + 1)) in ( divset (D1,j)) by A274, A238, A282, A285, A299, A303, INTEGRA2: 1;

                          then (D2 . (( indx (D2,D1,n1)) + 1)) in (( rng D) /\ ( divset (D1,j))) by A309, XBOOLE_0:def 4;

                          then (D2 . (( indx (D2,D1,n1)) + 1)) = (D2 . (( indx (D2,D1,n1)) + 2)) by A11, A236, A300, Th5;

                          hence contradiction by A289, A302, A291, SEQM_3:def 1;

                        end;

                        

                         A310: (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j)) implies (( indx (D2,D1,n1)) + 2) = ( indx (D2,D1,j))

                        proof

                          assume (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j));

                          then

                           A311: ((( indx (D2,D1,n1)) + 1) + 1) <= ( indx (D2,D1,j)) by NAT_1: 13;

                          (( indx (D2,D1,n1)) + 2) >= ( indx (D2,D1,j)) by A283, XREAL_1: 20;

                          hence thesis by A311, XXREAL_0: 1;

                        end;

                        

                         A312: 1 <= (( indx (D2,D1,n1)) + 1) by NAT_1: 12;

                        

                         A313: ( indx (D2,D1,j)) <= ( len LVf(D2)) by A263, INTEGRA1:def 7;

                        (D1 . n1) < (D1 . j) by A236, A241, A245, SEQM_3:def 1;

                        then

                         A314: ( indx (D2,D1,n1)) < ( indx (D2,D1,j)) by A259, A274, A261, A238, SEQ_4: 137;

                        then

                         A315: (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,j)) by NAT_1: 13;

                        then (( indx (D2,D1,n1)) + 1) <= ( len D2) by A263, XXREAL_0: 2;

                        then (( indx (D2,D1,n1)) + 1) <= ( len LVf(D2)) by INTEGRA1:def 7;

                        

                        then

                         A316: ( len ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) = ((( indx (D2,D1,j)) -' (( indx (D2,D1,n1)) + 1)) + 1) by A262, A315, A312, A313, FINSEQ_6: 118

                        .= ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A315, XREAL_1: 233

                        .= (( indx (D2,D1,j)) - ( indx (D2,D1,n1)));

                        (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,j)) by A314, NAT_1: 13;

                        then

                         A317: (( indx (D2,D1,n1)) + 1) = ( indx (D2,D1,j)) or (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j)) by XXREAL_0: 1;

                        

                         A318: ( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) <= (( upper_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1)))))

                        proof

                          per cases by A317, A310;

                            suppose

                             A319: (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) = 1;

                            

                             A320: (( indx (D2,D1,n1)) + 1) > 1 by A260, NAT_1: 13;

                            then ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . (( indx (D2,D1,n1)) + 1)) by A261, A319, INTEGRA1:def 4;

                            then

                             A321: ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D1 . j) by A13, A236, A319, INTEGRA1:def 19;

                            ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . ((( indx (D2,D1,n1)) + 1) - 1)) by A261, A319, A320, INTEGRA1:def 4;

                            then

                             A322: ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D1 . n1) by A13, A241, INTEGRA1:def 19;

                            ( lower_bound ( divset (D1,(n1 + 1)))) = (D1 . ((n1 + 1) - 1)) by A245, A280, A268, A282, INTEGRA1:def 4;

                            then

                             A323: ( divset (D2,(( indx (D2,D1,n1)) + 1))) = ( divset (D1,(n1 + 1))) by A245, A280, A268, A282, A322, A321, INTEGRA1:def 4;

                            

                             A324: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1)))) >= 0 by INTEGRA1: 9;

                            reconsider LV = ( LVf(D2) . (( indx (D2,D1,n1)) + 1)) as Element of REAL by XREAL_0:def 1;

                            1 = ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A319;

                            

                            then (( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . 1) = ( LVf(D2) . ((1 + (( indx (D2,D1,n1)) + 1)) - 1)) by A312, A313, FINSEQ_6: 122

                            .= ( LVf(D2) . (( indx (D2,D1,n1)) + 1));

                            then

                             A325: ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) = <*LV*> by A316, A319, FINSEQ_1: 40;

                            ( LVf(D2) . (( indx (D2,D1,n1)) + 1)) = (( lower_bound ( rng (f | ( divset (D2,(( indx (D2,D1,n1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) by A261, A319, INTEGRA1:def 7;

                            then ( LVf(D2) . (( indx (D2,D1,n1)) + 1)) <= (( upper_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1))))) by A1, A261, A319, A323, A324, Th18, XREAL_1: 64;

                            hence thesis by A325, FINSOP_1: 11;

                          end;

                            suppose

                             A326: (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) = 2;

                            (( indx (D2,D1,n1)) + 2) >= (2 + 1) by A260, XREAL_1: 6;

                            then

                             A327: (( indx (D2,D1,n1)) + 2) <> 1;

                            then

                             A328: ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 2)))) = (D2 . ( indx (D2,D1,j))) by A261, A326, INTEGRA1:def 4;

                            ((( indx (D2,D1,n1)) + 2) - 1) = (( indx (D2,D1,n1)) + 1);

                            then ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 2)))) = (D2 . (( indx (D2,D1,n1)) + 1)) by A261, A326, A327, INTEGRA1:def 4;

                            then

                             A329: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2)))) = ((D1 . j) - (D2 . (( indx (D2,D1,n1)) + 1))) by A238, A328, INTEGRA1:def 5;

                            

                             A330: ( upper_bound ( divset (D1,(n1 + 1)))) = (D1 . (n1 + 1)) by A245, A280, A268, A282, INTEGRA1:def 4;

                            ( lower_bound ( divset (D1,(n1 + 1)))) = (D1 . ((n1 + 1) - 1)) by A245, A280, A268, A282, INTEGRA1:def 4;

                            then

                             A331: ( vol ( divset (D1,(n1 + 1)))) = ((D1 . (n1 + 1)) - (D1 . n1)) by A330, INTEGRA1:def 5;

                            

                             A332: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2)))) >= 0 by INTEGRA1: 9;

                            

                             A333: ( indx (D2,D1,j)) <= ( len LVf(D2)) by A263, INTEGRA1:def 7;

                            

                             A334: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1)))) >= 0 by INTEGRA1: 9;

                            

                             A335: 1 <= (( indx (D2,D1,n1)) + 1) by NAT_1: 12;

                            

                             A336: (( indx (D2,D1,n1)) + 1) <= (( indx (D2,D1,n1)) + 2) by XREAL_1: 6;

                            then (( indx (D2,D1,n1)) + 1) <= ( len D2) by A263, A326, XXREAL_0: 2;

                            then

                             A337: (( indx (D2,D1,n1)) + 1) in ( dom D2) by A335, FINSEQ_3: 25;

                            then ( LVf(D2) . (( indx (D2,D1,n1)) + 1)) = (( lower_bound ( rng (f | ( divset (D2,(( indx (D2,D1,n1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) by INTEGRA1:def 7;

                            then

                             A338: ( LVf(D2) . (( indx (D2,D1,n1)) + 1)) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) by A1, A337, A334, Th18, XREAL_1: 64;

                            ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) = (1 + 1) by A326;

                            

                            then

                             A339: (( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . 2) = ( LVf(D2) . ((2 + (( indx (D2,D1,n1)) + 1)) - 1)) by A335, A336, A333, FINSEQ_6: 122

                            .= ( LVf(D2) . ((( indx (D2,D1,n1)) + 0 ) + 2));

                            ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) >= 1 by A326;

                            

                            then (( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . 1) = ( LVf(D2) . ((1 + (( indx (D2,D1,n1)) + 1)) - 1)) by A326, A335, A336, A333, FINSEQ_6: 122

                            .= ( LVf(D2) . (( indx (D2,D1,n1)) + 1));

                            then ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) = <*( LVf(D2) . (( indx (D2,D1,n1)) + 1)), ( LVf(D2) . (( indx (D2,D1,n1)) + 2))*> by A316, A326, A339, FINSEQ_1: 44;

                            then

                             A340: ( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) = (( LVf(D2) . (( indx (D2,D1,n1)) + 1)) + ( LVf(D2) . (( indx (D2,D1,n1)) + 2))) by RVSUM_1: 77;

                            

                             A341: (( indx (D2,D1,n1)) + 1) > 1 by A260, NAT_1: 13;

                            then

                             A342: ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . (( indx (D2,D1,n1)) + 1)) by A337, INTEGRA1:def 4;

                            ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . ((( indx (D2,D1,n1)) + 1) - 1)) by A337, A341, INTEGRA1:def 4;

                            then

                             A343: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = ((D2 . (( indx (D2,D1,n1)) + 1)) - (D1 . n1)) by A274, A342, INTEGRA1:def 5;

                            ( LVf(D2) . (( indx (D2,D1,n1)) + 2)) = (( lower_bound ( rng (f | ( divset (D2,(( indx (D2,D1,n1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2))))) by A261, A326, INTEGRA1:def 7;

                            then ( LVf(D2) . (( indx (D2,D1,n1)) + 2)) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2))))) by A1, A261, A326, A332, Th18, XREAL_1: 64;

                            then ( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) <= ((( upper_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) + (( upper_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2)))))) by A340, A338, XREAL_1: 7;

                            hence thesis by A282, A343, A329, A331;

                          end;

                        end;

                        

                         A344: (n1 + 1) <= ( len LVf(D1)) by A267, INTEGRA1:def 7;

                        

                        then

                         A345: ( len ( mid ( LVf(D1),(n1 + 1),j))) = ((j -' (n1 + 1)) + 1) by A244, A282, FINSEQ_6: 118

                        .= ((j - j) + 1) by A282, XREAL_1: 233

                        .= 1;

                        reconsider lv = (( lower_bound ( rng (f | ( divset (D1,(n1 + 1)))))) * ( vol ( divset (D1,(n1 + 1))))) as Element of REAL by XREAL_0:def 1;

                        ((n1 + 1) + 1) <= (j + 1) by A257, XREAL_1: 6;

                        then 1 <= ((j + 1) - (n1 + 1)) by XREAL_1: 19;

                        

                        then (( mid ( LVf(D1),(n1 + 1),j)) . 1) = ( LVf(D1) . ((1 - 1) + (n1 + 1))) by A244, A282, A344, FINSEQ_6: 122

                        .= (( lower_bound ( rng (f | ( divset (D1,(n1 + 1)))))) * ( vol ( divset (D1,(n1 + 1))))) by A268, INTEGRA1:def 7;

                        then ( mid ( LVf(D1),(n1 + 1),j)) = <*lv*> by A345, FINSEQ_1: 40;

                        then

                         A346: ( Sum ( mid ( LVf(D1),(n1 + 1),j))) = (( lower_bound ( rng (f | ( divset (D1,(n1 + 1)))))) * ( vol ( divset (D1,(n1 + 1))))) by FINSOP_1: 11;

                        ( divset (D1,(n1 + 1))) c= A by A268, INTEGRA1: 8;

                        then

                         A347: ( lower_bound ( rng (f | ( divset (D1,(n1 + 1)))))) >= ( lower_bound ( rng f)) by A1, Lm4;

                        (n1 + 1) in ( Seg ( len D1)) by A268, FINSEQ_1:def 3;

                        then (n1 + 1) in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by INTEGRA1:def 6;

                        then

                         A348: (n1 + 1) in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

                        ( vol ( divset (D1,(n1 + 1)))) = (( upper_volume (( chi (A,A)),D1)) . (n1 + 1)) by A268, INTEGRA1: 20;

                        then ( vol ( divset (D1,(n1 + 1)))) in ( rng ( upper_volume (( chi (A,A)),D1))) by A348, FUNCT_1:def 3;

                        then

                         A349: ( vol ( divset (D1,(n1 + 1)))) <= ( delta D1) by XXREAL_2:def 8;

                        (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                        then

                         A350: ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( vol ( divset (D1,(n1 + 1))))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1)) by A349, XREAL_1: 64;

                        ( vol ( divset (D1,(n1 + 1)))) >= 0 by INTEGRA1: 9;

                        then ( Sum ( mid ( LVf(D1),(n1 + 1),j))) >= (( lower_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1))))) by A346, A347, XREAL_1: 64;

                        then (( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) - ( Sum ( mid ( LVf(D1),(n1 + 1),j)))) <= ((( upper_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1))))) - (( lower_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1)))))) by A318, XREAL_1: 13;

                        hence thesis by A350, XXREAL_0: 2;

                      end;

                        suppose

                         A351: (n1 + 1) < j;

                        

                         A352: n1 < (n1 + 1) by NAT_1: 13;

                        then

                         A353: (D1 . n1) < (D1 . (n1 + 1)) by A241, A268, SEQM_3:def 1;

                        then

                        consider B be non empty closed_interval Subset of REAL , MD1,MD2 be Division of B such that

                         A354: (D1 . n1) = ( lower_bound B) and ( upper_bound B) = (MD2 . ( len MD2)) and

                         A355: ( upper_bound B) = (MD1 . ( len MD1)) and

                         A356: MD1 <= MD2 and

                         A357: MD1 = ( mid (D1,(n1 + 1),j)) and

                         A358: MD2 = ( mid (D2,( indx (D2,D1,(n1 + 1))),( indx (D2,D1,j)))) by A13, A236, A257, A268, A273, Th15;

                        

                         A359: ( delta MD1) >= 0 by Th9;

                        

                         A360: ( len MD1) = ((j -' (n1 + 1)) + 1) by A257, A265, A266, A244, A267, A357, FINSEQ_6: 118;

                        

                        then

                         A361: ((( len MD1) + (n1 + 1)) - 1) = ((((j - (n1 + 1)) + 1) + (n1 + 1)) - 1) by A257, XREAL_1: 233

                        .= j;

                        (j -' (n1 + 1)) = (j - (n1 + 1)) by A257, XREAL_1: 233;

                        then

                         A362: ((j -' (n1 + 1)) + 1) = (j - n1);

                        then

                         A363: ( len MD1) = (j - n1) by A257, A265, A266, A244, A267, A357, FINSEQ_6: 118;

                        

                         A364: B c= A

                        proof

                          let x1 be object;

                          

                           A365: ( rng D1) c= A by INTEGRA1:def 2;

                          (D1 . n1) in ( rng D1) by A241, FUNCT_1:def 3;

                          then

                           A366: ( lower_bound A) <= (D1 . n1) by A365, INTEGRA2: 1;

                          assume

                           A367: x1 in B;

                          then

                          reconsider x1 as Real;

                          

                           A368: x1 <= (MD1 . ( len MD1)) by A355, A367, INTEGRA2: 1;

                          (D1 . j) in ( rng D1) by A236, FUNCT_1:def 3;

                          then

                           A369: (D1 . j) <= ( upper_bound A) by A365, INTEGRA2: 1;

                          (D1 . n1) <= x1 by A354, A367, INTEGRA2: 1;

                          then

                           A370: ( lower_bound A) <= x1 by A366, XXREAL_0: 2;

                          (MD1 . ( len MD1)) = (D1 . (((j - n1) - 1) + (n1 + 1))) by A257, A281, A266, A244, A357, A362, A363, FINSEQ_6: 122

                          .= (D1 . j);

                          then x1 <= ( upper_bound A) by A368, A369, XXREAL_0: 2;

                          hence thesis by A370, INTEGRA2: 1;

                        end;

                        then

                        reconsider g = (f | B) as Function of B, REAL by FUNCT_2: 32;

                        

                         A371: ( len ( lower_volume (g,MD1))) = ( len MD1) by INTEGRA1:def 7

                        .= ((j -' (n1 + 1)) + 1) by A257, A265, A266, A244, A267, A357, FINSEQ_6: 118

                        .= ((j - (n1 + 1)) + 1) by A257, XREAL_1: 233;

                        

                         A372: ( len MD1) in ( dom MD1) by FINSEQ_5: 6;

                        then

                         A373: 1 <= ( len MD1) by FINSEQ_3: 25;

                        

                         A374: ( lower_bound ( divset (MD1,( len MD1)))) = ( lower_bound ( divset (D1,j))) & ( upper_bound ( divset (MD1,( len MD1)))) = ( upper_bound ( divset (D1,j)))

                        proof

                          per cases ;

                            suppose

                             A375: ( len MD1) = 1;

                            then

                             A376: ( upper_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len MD1)) by A372, INTEGRA1:def 4;

                            

                             A377: ( upper_bound ( divset (D1,j))) = (D1 . j) by A236, A245, A280, INTEGRA1:def 4;

                            ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A236, A245, A280, INTEGRA1:def 4;

                            hence thesis by A265, A266, A354, A357, A361, A372, A375, A376, A377, FINSEQ_6: 118, INTEGRA1:def 4;

                          end;

                            suppose

                             A378: ( len MD1) <> 1;

                            then (( len MD1) - 1) in ( dom MD1) by A372, INTEGRA1: 7;

                            then

                             A379: (( len MD1) - 1) >= 1 by FINSEQ_3: 25;

                            ( len MD1) <= (( len MD1) + 1) by NAT_1: 11;

                            then

                             A380: (( len MD1) - 1) <= ( len MD1) by XREAL_1: 20;

                            ( upper_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len MD1)) by A372, A378, INTEGRA1:def 4;

                            then

                             A381: ( upper_bound ( divset (MD1,( len MD1)))) = (D1 . j) by A257, A266, A244, A357, A360, A361, A373, FINSEQ_6: 122;

                            

                             A382: (((( len MD1) - 1) + (n1 + 1)) - 1) = (j - 1) by A363;

                            ( lower_bound ( divset (MD1,( len MD1)))) = (MD1 . (( len MD1) - 1)) by A372, A378, INTEGRA1:def 4;

                            then ( lower_bound ( divset (MD1,( len MD1)))) = (D1 . (j - 1)) by A257, A266, A244, A357, A360, A382, A379, A380, FINSEQ_6: 122;

                            hence thesis by A236, A245, A280, A381, INTEGRA1:def 4;

                          end;

                        end;

                        

                         A383: ( len MD1) in ( dom MD1) by FINSEQ_5: 6;

                        

                         A384: ( upper_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len MD1))

                        proof

                          per cases ;

                            suppose ( len MD1) = 1;

                            hence thesis by A383, INTEGRA1:def 4;

                          end;

                            suppose ( len MD1) <> 1;

                            hence thesis by A383, INTEGRA1:def 4;

                          end;

                        end;

                        (D1 . n1) < (D1 . (n1 + 1)) by A241, A268, A352, SEQM_3:def 1;

                        then ( indx (D2,D1,n1)) < ( indx (D2,D1,(n1 + 1))) by A259, A274, A269, A271, SEQ_4: 137;

                        then

                         A385: (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,(n1 + 1))) by NAT_1: 13;

                        then

                         A386: (( indx (D2,D1,n1)) + 1) <= ( len D2) by A277, XXREAL_0: 2;

                        ( vol B) = (( upper_bound B) - (D1 . n1)) by A354, INTEGRA1:def 5;

                        then ( vol B) = ((D1 . j) - (D1 . n1)) by A236, A245, A280, A355, A374, A384, INTEGRA1:def 4;

                        then

                         A387: ( vol B) <> 0 by A236, A241, A245, SEQM_3:def 1;

                        

                         A388: 1 <= (( indx (D2,D1,n1)) + 1) by A260, NAT_1: 13;

                        

                         A389: ( indx (D2,D1,n1)) < (( indx (D2,D1,n1)) + 1) by NAT_1: 13;

                        

                         A390: ( indx (D2,D1,(n1 + 1))) = (( indx (D2,D1,n1)) + 1)

                        proof

                          assume ( indx (D2,D1,(n1 + 1))) <> (( indx (D2,D1,n1)) + 1);

                          then

                           A391: ( indx (D2,D1,(n1 + 1))) > (( indx (D2,D1,n1)) + 1) by A385, XXREAL_0: 1;

                          

                           A392: (( indx (D2,D1,n1)) + 1) in ( dom D2) by A388, A386, FINSEQ_3: 25;

                          then

                           A393: (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D2) by FUNCT_1:def 3;

                          now

                            per cases by A14, A393, XBOOLE_0:def 3;

                              suppose (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D1);

                              then

                              consider n2 be Element of NAT such that

                               A394: n2 in ( dom D1) and

                               A395: (D2 . (( indx (D2,D1,n1)) + 1)) = (D1 . n2) by PARTFUN1: 3;

                              (D2 . ( indx (D2,D1,n1))) < (D2 . (( indx (D2,D1,n1)) + 1)) by A259, A389, A392, SEQM_3:def 1;

                              then n1 < n2 by A241, A274, A394, A395, SEQ_4: 137;

                              then

                               A396: (n1 + 1) <= n2 by NAT_1: 13;

                              (D1 . n2) < (D1 . (n1 + 1)) by A269, A271, A391, A392, A395, SEQM_3:def 1;

                              hence contradiction by A268, A394, A396, SEQ_4: 137;

                            end;

                              suppose

                               A397: (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D);

                              

                               A398: (D . i) <= ( upper_bound ( divset (D1,n1))) by A242, INTEGRA2: 1;

                              

                               A399: ( upper_bound ( divset (D1,n1))) = (D1 . n1)

                              proof

                                per cases ;

                                  suppose n1 = 1;

                                  hence thesis by A241, INTEGRA1:def 4;

                                end;

                                  suppose n1 <> 1;

                                  hence thesis by A241, INTEGRA1:def 4;

                                end;

                              end;

                              consider n2 be Element of NAT such that

                               A400: n2 in ( dom D) and

                               A401: (D2 . (( indx (D2,D1,n1)) + 1)) = (D . n2) by A397, PARTFUN1: 3;

                              (D1 . n1) < (D . n2) by A259, A274, A389, A392, A401, SEQM_3:def 1;

                              then (D . i) < (D . n2) by A398, A399, XXREAL_0: 2;

                              then i < n2 by A240, A400, SEQ_4: 137;

                              then

                               A402: (i + 1) <= n2 by NAT_1: 13;

                              ((n1 + 1) + 1) <= j by A351, NAT_1: 13;

                              then

                               A403: (n1 + 1) <= (j - 1) by XREAL_1: 19;

                              (j - 1) in ( dom D1) by A236, A245, A280, INTEGRA1: 7;

                              then

                               A404: (D1 . (n1 + 1)) <= (D1 . (j - 1)) by A268, A403, SEQ_4: 137;

                              

                               A405: ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A237, INTEGRA2: 1;

                              ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A236, A245, A280, INTEGRA1:def 4;

                              then

                               A406: (D1 . (n1 + 1)) <= (D . (i + 1)) by A404, A405, XXREAL_0: 2;

                              (D . n2) < (D1 . (n1 + 1)) by A269, A271, A391, A392, A401, SEQM_3:def 1;

                              then (D . n2) < (D . (i + 1)) by A406, XXREAL_0: 2;

                              hence contradiction by A235, A400, A402, SEQ_4: 137;

                            end;

                          end;

                          hence contradiction;

                        end;

                        

                         A407: j <= ( len LVf(D1)) by A266, INTEGRA1:def 7;

                        

                         A408: for k be Nat st 1 <= k & k <= ( len ( lower_volume (g,MD1))) holds (( lower_volume (g,MD1)) . k) = (( mid ( LVf(D1),(n1 + 1),j)) . k)

                        proof

                          let k be Nat;

                          assume that

                           A409: 1 <= k and

                           A410: k <= ( len ( lower_volume (g,MD1)));

                          k in ( Seg ( len ( lower_volume (g,MD1)))) by A409, A410, FINSEQ_1: 1;

                          then

                           A411: k in ( Seg ( len MD1)) by INTEGRA1:def 7;

                          then

                           A412: k in ( dom MD1) by FINSEQ_1:def 3;

                          k in ( dom MD1) by A411, FINSEQ_1:def 3;

                          then

                           A413: (( lower_volume (g,MD1)) . k) = (( lower_bound ( rng (g | ( divset (MD1,k))))) * ( vol ( divset (MD1,k)))) by INTEGRA1:def 7;

                          consider k2 be Element of NAT such that

                           A414: (n1 + 1) = (1 + k2);

                          

                           A415: 1 <= (k + k2) by A409, NAT_1: 12;

                          k <= (j - ((n1 + 1) - 1)) by A371, A410;

                          then (k + ((n1 + 1) - 1)) <= j by XREAL_1: 19;

                          then (k + k2) <= ( len D1) by A266, A414, XXREAL_0: 2;

                          then

                           A416: (k + k2) in ( Seg ( len D1)) by A415, FINSEQ_1: 1;

                          then

                           A417: (k + k2) in ( dom D1) by FINSEQ_1:def 3;

                          (1 + 1) <= (k + k2) by A258, A409, A414, XREAL_1: 7;

                          then

                           A418: 1 < (k + k2) by NAT_1: 13;

                          

                           A419: k2 = ((n1 + 1) - 1) by A414;

                          

                           A420: ( lower_bound ( divset (D1,(k + k2)))) = ( lower_bound ( divset (MD1,k))) & ( upper_bound ( divset (D1,(k + k2)))) = ( upper_bound ( divset (MD1,k)))

                          proof

                            per cases ;

                              suppose

                               A421: k = 1;

                              then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A412, INTEGRA1:def 4;

                              then

                               A422: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A257, A266, A244, A357, A371, A409, A410, FINSEQ_6: 122;

                              ( lower_bound ( divset (MD1,k))) = (D1 . n1) by A354, A412, A421, INTEGRA1:def 4;

                              hence thesis by A419, A418, A417, A421, A422, INTEGRA1:def 4;

                            end;

                              suppose

                               A423: k <> 1;

                              then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A412, INTEGRA1:def 4;

                              then

                               A424: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A257, A266, A244, A357, A371, A409, A410, FINSEQ_6: 122;

                              

                               A425: (k - 1) <= ((j - (n1 + 1)) + 1) by A371, A410, XREAL_1: 146, XXREAL_0: 2;

                              

                               A426: ( lower_bound ( divset (MD1,k))) = (MD1 . (k - 1)) by A412, A423, INTEGRA1:def 4;

                              

                               A427: (k - 1) in ( dom MD1) by A412, A423, INTEGRA1: 7;

                              then 1 <= (k - 1) by FINSEQ_3: 25;

                              then ( lower_bound ( divset (MD1,k))) = (D1 . (((k - 1) + (n1 + 1)) - 1)) by A257, A266, A244, A357, A427, A425, A426, FINSEQ_6: 122;

                              hence thesis by A414, A418, A417, A424, INTEGRA1:def 4;

                            end;

                          end;

                          ( divset (MD1,k)) = [.( lower_bound ( divset (MD1,k))), ( upper_bound ( divset (MD1,k))).] by INTEGRA1: 4;

                          then

                           A428: ( divset (D1,(k + k2))) = ( divset (MD1,k)) by A420, INTEGRA1: 4;

                          

                           A429: (k + k2) in ( dom D1) by A416, FINSEQ_1:def 3;

                          

                           A430: (( mid ( LVf(D1),(n1 + 1),j)) . k) = ( LVf(D1) . ((k + (n1 + 1)) - 1)) by A257, A244, A371, A407, A409, A410, FINSEQ_6: 122

                          .= (( lower_bound ( rng (f | ( divset (D1,(k + k2)))))) * ( vol ( divset (D1,(k + k2))))) by A414, A429, INTEGRA1:def 7;

                          k in ( dom MD1) by A411, FINSEQ_1:def 3;

                          then ( divset (D1,(k + k2))) c= B by A428, INTEGRA1: 8;

                          hence thesis by A413, A430, A428, FUNCT_1: 51;

                        end;

                        

                         A431: (g | B) is bounded

                        proof

                          consider a be Real such that

                           A432: for x be object st x in (A /\ ( dom f)) holds a <= (f . x) by A1, RFUNCT_1: 71;

                          for x be object st x in (B /\ ( dom g)) holds a <= (g . x)

                          proof

                            let x be object;

                            

                             A433: (( dom f) /\ B) c= (( dom f) /\ A) by A364, XBOOLE_1: 26;

                            assume x in (B /\ ( dom g));

                            then

                             A434: x in ( dom g) by XBOOLE_0:def 4;

                            then x in (( dom f) /\ B) by RELAT_1: 61;

                            then a <= (f . x) by A432, A433;

                            hence thesis by A434, FUNCT_1: 47;

                          end;

                          then

                           A435: (g | B) is bounded_below by RFUNCT_1: 71;

                          consider a be Real such that

                           A436: for x be object st x in (A /\ ( dom f)) holds (f . x) <= a by A1, RFUNCT_1: 70;

                          for x be object st x in (B /\ ( dom g)) holds (g . x) <= a

                          proof

                            let x be object;

                            

                             A437: (( dom f) /\ B) c= (( dom f) /\ A) by A364, XBOOLE_1: 26;

                            assume x in (B /\ ( dom g));

                            then

                             A438: x in ( dom g) by XBOOLE_0:def 4;

                            then x in (( dom f) /\ B) by RELAT_1: 61;

                            then a >= (f . x) by A436, A437;

                            hence thesis by A438, FUNCT_1: 47;

                          end;

                          then (g | B) is bounded_above by RFUNCT_1: 70;

                          hence thesis by A435;

                        end;

                        ( rng f) is bounded_below by A1, INTEGRA1: 11;

                        then

                         A439: ( lower_bound ( rng f)) <= ( lower_bound ( rng g)) by RELAT_1: 70, SEQ_4: 47;

                        ( rng f) is bounded_above by A1, INTEGRA1: 13;

                        then ( upper_bound ( rng f)) >= ( upper_bound ( rng g)) by RELAT_1: 70, SEQ_4: 48;

                        then (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= (( upper_bound ( rng g)) - ( lower_bound ( rng g))) by A439, XREAL_1: 13;

                        then

                         A440: ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) >= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta MD1)) by A359, XREAL_1: 64;

                        

                         A441: n1 < (j - 1) by A351, XREAL_1: 20;

                        

                         A442: ( indx (D2,D1,j)) <= ( len LVf(D2)) by A263, INTEGRA1:def 7;

                        

                         A443: ( len MD2) = ((( indx (D2,D1,j)) -' ( indx (D2,D1,(n1 + 1)))) + 1) by A272, A270, A277, A262, A263, A358, FINSEQ_6: 118;

                        then

                         A444: ( len MD2) = ((( indx (D2,D1,j)) - ( indx (D2,D1,(n1 + 1)))) + 1) by A272, XREAL_1: 233;

                        then

                         A445: ( len ( lower_volume (g,MD2))) = ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A390, INTEGRA1:def 7;

                        for x1 be object holds x1 in (( rng MD1) \/ {(D . (i + 1))}) implies x1 in ( rng MD2)

                        proof

                          let x1 be object;

                          assume

                           A446: x1 in (( rng MD1) \/ {(D . (i + 1))});

                          then

                          reconsider x1 as Real;

                          now

                            per cases by A446, XBOOLE_0:def 3;

                              suppose

                               A447: x1 in ( rng MD1);

                              ( rng MD1) <> {} ;

                              then 1 in ( dom MD1) by FINSEQ_3: 32;

                              then

                               A448: 1 <= ( len MD1) by FINSEQ_3: 25;

                              ( rng MD1) c= ( rng D1) by A357, FINSEQ_6: 119;

                              then

                               A449: x1 in ( rng D1) by A447;

                              ( rng D1) c= ( rng D2) by A13, INTEGRA1:def 18;

                              then

                              consider k such that

                               A450: k in ( dom D2) and

                               A451: (D2 . k) = x1 by A449, PARTFUN1: 3;

                              (MD1 . 1) = (D1 . (n1 + 1)) by A265, A266, A244, A267, A357, FINSEQ_6: 118;

                              then (D2 . ( indx (D2,D1,(n1 + 1)))) <= x1 by A271, A447, Th16;

                              then

                               A452: ( indx (D2,D1,(n1 + 1))) <= k by A269, A450, A451, SEQM_3:def 1;

                              then

                              consider n be Nat such that

                               A453: (k + 1) = (( indx (D2,D1,(n1 + 1))) + n) by NAT_1: 10, NAT_1: 12;

                              

                               A454: ( len MD1) = ((j -' (n1 + 1)) + 1) by A257, A265, A266, A244, A267, A357, FINSEQ_6: 118;

                              

                              then ((( len MD1) + (n1 + 1)) - 1) = ((((j - (n1 + 1)) + 1) + (n1 + 1)) - 1) by A257, XREAL_1: 233

                              .= j;

                              then (MD1 . ( len MD1)) = (D1 . j) by A257, A266, A244, A357, A448, A454, FINSEQ_6: 122;

                              then x1 <= (D2 . ( indx (D2,D1,j))) by A238, A447, Th16;

                              then k <= ( indx (D2,D1,j)) by A261, A450, A451, SEQM_3:def 1;

                              then (k - ( indx (D2,D1,(n1 + 1)))) <= (( indx (D2,D1,j)) - ( indx (D2,D1,(n1 + 1)))) by XREAL_1: 9;

                              then

                               A455: ((k - ( indx (D2,D1,(n1 + 1)))) + 1) <= ((( indx (D2,D1,j)) - ( indx (D2,D1,(n1 + 1)))) + 1) by XREAL_1: 6;

                              (( indx (D2,D1,(n1 + 1))) + 1) <= (k + 1) by A452, XREAL_1: 6;

                              then

                               A456: 1 <= ((k + 1) - ( indx (D2,D1,(n1 + 1)))) by XREAL_1: 19;

                              then

                               A457: n in ( dom MD2) by A444, A455, A453, FINSEQ_3: 25;

                              (MD2 . n) = (D2 . ((n + ( indx (D2,D1,(n1 + 1)))) - 1)) by A272, A270, A263, A358, A456, A455, A453, FINSEQ_6: 122

                              .= (D2 . k) by A453;

                              hence x1 in ( rng MD2) by A451, A457, FUNCT_1:def 3;

                            end;

                              suppose x1 in {(D . (i + 1))};

                              then

                               A458: x1 = (D . (i + 1)) by TARSKI:def 1;

                              reconsider j1 = (j - 1) as Element of NAT by A236, A245, A280, INTEGRA1: 7;

                              

                               A459: ( rng D) c= ( rng D2) by A12, INTEGRA1:def 18;

                              (D . (i + 1)) in ( rng D) by A235, FUNCT_1:def 3;

                              then

                              consider k such that

                               A460: k in ( dom D2) and

                               A461: x1 = (D2 . k) by A458, A459, PARTFUN1: 3;

                              (D . (i + 1)) <= ( upper_bound ( divset (D1,j))) by A237, INTEGRA2: 1;

                              then x1 <= (D1 . j) by A236, A245, A280, A458, INTEGRA1:def 4;

                              then

                               A462: (D2 . k) <= (D2 . ( indx (D2,D1,j))) by A13, A236, A461, INTEGRA1:def 19;

                              n1 < j1 by A351, XREAL_1: 20;

                              then

                               A463: (n1 + 1) <= j1 by NAT_1: 13;

                              (j - 1) in ( dom D1) by A236, A245, A280, INTEGRA1: 7;

                              then

                               A464: (D1 . (n1 + 1)) <= (D1 . (j - 1)) by A268, A463, SEQ_4: 137;

                              ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A237, INTEGRA2: 1;

                              then (D1 . (j - 1)) <= x1 by A236, A245, A280, A458, INTEGRA1:def 4;

                              then (D2 . ( indx (D2,D1,(n1 + 1)))) <= (D2 . k) by A271, A461, A464, XXREAL_0: 2;

                              hence x1 in ( rng MD2) by A269, A261, A358, A460, A461, A462, Th17;

                            end;

                          end;

                          hence thesis;

                        end;

                        then

                         A465: (( rng MD1) \/ {(D . (i + 1))}) c= ( rng MD2);

                        ( rng MD2) <> {} ;

                        then 1 in ( dom MD2) by FINSEQ_3: 32;

                        then

                         A466: 1 <= ( len MD2) by FINSEQ_3: 25;

                        

                         A467: ((( len MD2) - 1) + ( indx (D2,D1,(n1 + 1)))) = ( indx (D2,D1,j)) by A444;

                        for x1 be object holds x1 in ( rng MD2) implies x1 in (( rng MD1) \/ {(D . (i + 1))})

                        proof

                          let x1 be object;

                          assume

                           A468: x1 in ( rng MD2);

                          then

                          reconsider x1 as Real;

                          (MD2 . 1) = (D2 . ( indx (D2,D1,(n1 + 1)))) by A270, A277, A262, A263, A358, FINSEQ_6: 118;

                          then

                           A469: (D1 . (n1 + 1)) <= x1 by A271, A468, Th16;

                          (MD2 . ( len MD2)) = (D2 . ( indx (D2,D1,j))) by A272, A270, A263, A358, A466, A443, A467, FINSEQ_6: 122;

                          then

                           A470: x1 <= (D1 . j) by A238, A468, Th16;

                          

                           A471: ( rng MD2) c= ( rng D2) by A358, FINSEQ_6: 119;

                          now

                            per cases by A14, A468, A471, XBOOLE_0:def 3;

                              suppose x1 in ( rng D1);

                              then

                              consider k such that

                               A472: k in ( dom D1) and

                               A473: (D1 . k) = x1 by PARTFUN1: 3;

                              

                               A474: (n1 + 1) <= k by A268, A469, A472, A473, SEQM_3:def 1;

                              then

                               A475: 1 <= (k - n1) by XREAL_1: 19;

                              n1 <= (n1 + 1) by NAT_1: 11;

                              then

                              consider n be Nat such that

                               A476: k = (n1 + n) by A474, NAT_1: 10, XXREAL_0: 2;

                              

                               A477: k <= j by A236, A470, A472, A473, SEQM_3:def 1;

                              then (k - n1) <= ( len MD1) by A363, XREAL_1: 9;

                              then n in ( dom MD1) by A475, A476, FINSEQ_3: 25;

                              then

                               A478: (MD1 . n) in ( rng MD1) by FUNCT_1:def 3;

                              ((j - (n1 + 1)) + 1) = (j - n1);

                              then

                               A479: (k - n1) <= ((j - (n1 + 1)) + 1) by A477, XREAL_1: 9;

                              (MD1 . n) = (D1 . (((k - n1) - 1) + (n1 + 1))) by A257, A266, A244, A357, A475, A479, A476, FINSEQ_6: 122

                              .= (D1 . k);

                              hence x1 in (( rng MD1) \/ {(D . (i + 1))}) by A473, A478, XBOOLE_0:def 3;

                            end;

                              suppose x1 in ( rng D);

                              then

                              consider n such that

                               A480: n in ( dom D) and

                               A481: (D . n) = x1 by PARTFUN1: 3;

                              

                               A482: not (i + 1) < n

                              proof

                                

                                 A483: ( upper_bound ( divset (D1,j))) = (D1 . j)

                                proof

                                  per cases ;

                                    suppose j = 1;

                                    hence thesis by A236, INTEGRA1:def 4;

                                  end;

                                    suppose j <> 1;

                                    hence thesis by A236, INTEGRA1:def 4;

                                  end;

                                end;

                                reconsider y1 = (D . (i + 1)) as Real;

                                

                                 A484: (D . n) in ( rng D) by A480, FUNCT_1:def 3;

                                assume (i + 1) < n;

                                then

                                 A485: (D . (i + 1)) < (D . n) by A235, A480, SEQM_3:def 1;

                                ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A237, INTEGRA2: 1;

                                then ( lower_bound ( divset (D1,j))) <= (D . n) by A485, XXREAL_0: 2;

                                then (D . n) in ( divset (D1,j)) by A470, A481, A483, INTEGRA2: 1;

                                then

                                 A486: x1 in (( rng D) /\ ( divset (D1,j))) by A481, A484, XBOOLE_0:def 4;

                                (D . (i + 1)) in ( rng D) by A235, FUNCT_1:def 3;

                                then y1 in (( rng D) /\ ( divset (D1,j))) by A237, XBOOLE_0:def 4;

                                hence contradiction by A11, A236, A481, A485, A486, Th5;

                              end;

                              

                               A487: ( upper_bound ( divset (D1,n1))) = (D1 . n1)

                              proof

                                per cases ;

                                  suppose n1 = 1;

                                  hence thesis by A241, INTEGRA1:def 4;

                                end;

                                  suppose n1 <> 1;

                                  hence thesis by A241, INTEGRA1:def 4;

                                end;

                              end;

                              (D . i) <= ( upper_bound ( divset (D1,n1))) by A242, INTEGRA2: 1;

                              then (D . i) < (D1 . (n1 + 1)) by A353, A487, XXREAL_0: 2;

                              then (D . i) < (D . n) by A469, A481, XXREAL_0: 2;

                              then i < n by A240, A480, SEQ_4: 137;

                              then (i + 1) <= n by NAT_1: 13;

                              then (i + 1) = n or (i + 1) < n by XXREAL_0: 1;

                              then x1 in {(D . (i + 1))} by A481, A482, TARSKI:def 1;

                              hence x1 in (( rng MD1) \/ {(D . (i + 1))}) by XBOOLE_0:def 3;

                            end;

                          end;

                          hence thesis;

                        end;

                        then ( rng MD2) c= (( rng MD1) \/ {(D . (i + 1))});

                        then

                         A488: ( rng MD2) = (( rng MD1) \/ {(D . (i + 1))}) by A465, XBOOLE_0:def 10;

                        ( delta MD1) in ( rng ( upper_volume (( chi (B,B)),MD1))) by XXREAL_2:def 8;

                        then

                        consider k such that

                         A489: k in ( dom ( upper_volume (( chi (B,B)),MD1))) and

                         A490: (( upper_volume (( chi (B,B)),MD1)) . k) = ( delta MD1) by PARTFUN1: 3;

                        

                         A491: k in ( Seg ( len ( upper_volume (( chi (B,B)),MD1)))) by A489, FINSEQ_1:def 3;

                        then

                         A492: k in ( Seg ( len MD1)) by INTEGRA1:def 6;

                        then

                         A493: k in ( dom MD1) by FINSEQ_1:def 3;

                        

                         A494: k <= ( len MD1) by A492, FINSEQ_1: 1;

                        then (k + n1) <= j by A363, XREAL_1: 19;

                        then

                         A495: (k + n1) <= ( len D1) by A266, XXREAL_0: 2;

                        

                         A496: 1 <= k by A491, FINSEQ_1: 1;

                        

                         A497: (n1 + 1) > 1 by A280, NAT_1: 13;

                        then n1 > (1 - 1) by XREAL_1: 19;

                        then

                         A498: k < (k + n1) by XREAL_1: 29;

                        then 1 < (k + n1) by A496, XXREAL_0: 2;

                        then

                         A499: (k + n1) in ( dom D1) by A495, FINSEQ_3: 25;

                        ( lower_bound ( divset (MD1,k))) = ( lower_bound ( divset (D1,(k + n1)))) & ( upper_bound ( divset (MD1,k))) = ( upper_bound ( divset (D1,(k + n1))))

                        proof

                          per cases ;

                            suppose

                             A500: k = 1;

                            then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A493, INTEGRA1:def 4;

                            then

                             A501: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A257, A266, A244, A357, A360, A496, A494, FINSEQ_6: 122;

                            ( lower_bound ( divset (D1,(k + n1)))) = (D1 . ((k + n1) - 1)) by A496, A498, A499, INTEGRA1:def 4;

                            hence thesis by A354, A497, A493, A499, A500, A501, INTEGRA1:def 4;

                          end;

                            suppose

                             A502: k <> 1;

                            then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A493, INTEGRA1:def 4;

                            then

                             A503: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A257, A266, A244, A357, A360, A496, A494, FINSEQ_6: 122;

                            

                             A504: ( lower_bound ( divset (MD1,k))) = (MD1 . (k - 1)) by A493, A502, INTEGRA1:def 4;

                            

                             A505: (k - 1) in ( dom MD1) by A493, A502, INTEGRA1: 7;

                            then

                             A506: (k - 1) <= ( len MD1) by FINSEQ_3: 25;

                            1 <= (k - 1) by A505, FINSEQ_3: 25;

                            then ( lower_bound ( divset (MD1,k))) = (D1 . (((k - 1) + (n1 + 1)) - 1)) by A257, A266, A244, A357, A360, A505, A506, A504, FINSEQ_6: 122;

                            hence thesis by A496, A498, A499, A503, INTEGRA1:def 4;

                          end;

                        end;

                        then ( divset (MD1,k)) = [.( lower_bound ( divset (D1,(k + n1)))), ( upper_bound ( divset (D1,(k + n1)))).] by INTEGRA1: 4;

                        then

                         A507: ( divset (MD1,k)) = ( divset (D1,(k + n1))) by INTEGRA1: 4;

                        (k + n1) in ( Seg ( len D1)) by A499, FINSEQ_1:def 3;

                        then (k + n1) in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by INTEGRA1:def 6;

                        then

                         A508: (k + n1) in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

                        k in ( dom MD1) by A492, FINSEQ_1:def 3;

                        then ( delta MD1) = ( vol ( divset (MD1,k))) by A490, INTEGRA1: 20;

                        then ( delta MD1) = (( upper_volume (( chi (A,A)),D1)) . (k + n1)) by A499, A507, INTEGRA1: 20;

                        then ( delta MD1) in ( rng ( upper_volume (( chi (A,A)),D1))) by A508, FUNCT_1:def 3;

                        then ( delta MD1) <= ( max ( rng ( upper_volume (( chi (A,A)),D1)))) by XXREAL_2:def 8;

                        then

                         A509: ( delta MD1) <= ( delta D1);

                        (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                        then

                         A510: ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1)) by A509, XREAL_1: 64;

                        ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A237, INTEGRA2: 1;

                        then

                         A511: (D1 . (j - 1)) <= (D . (i + 1)) by A236, A245, A280, INTEGRA1:def 4;

                        

                         A512: (D . (i + 1)) <= ( upper_bound ( divset (D1,j))) by A237, INTEGRA2: 1;

                        

                         A513: (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,j)) by A272, A385, XXREAL_0: 2;

                        

                         A514: for k be Nat st 1 <= k & k <= ( len ( lower_volume (g,MD2))) holds (( lower_volume (g,MD2)) . k) = (( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . k)

                        proof

                          let k be Nat;

                          assume that

                           A515: 1 <= k and

                           A516: k <= ( len ( lower_volume (g,MD2)));

                          

                           A517: k in ( Seg ( len ( lower_volume (g,MD2)))) by A515, A516, FINSEQ_1: 1;

                          

                           A518: (( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . k) = ( LVf(D2) . ((k + (( indx (D2,D1,n1)) + 1)) - 1)) by A388, A445, A442, A513, A515, A516, FINSEQ_6: 122;

                          1 <= (( indx (D2,D1,n1)) + 1) by NAT_1: 12;

                          then (1 + 1) <= (k + (( indx (D2,D1,n1)) + 1)) by A515, XREAL_1: 7;

                          then

                           A519: 1 <= ((k + (( indx (D2,D1,n1)) + 1)) - 1) by XREAL_1: 19;

                          consider k2 be Element of NAT such that

                           A520: (( indx (D2,D1,n1)) + 1) = (1 + k2);

                          k <= (( indx (D2,D1,j)) - ((( indx (D2,D1,n1)) + 1) - 1)) by A444, A390, A516, INTEGRA1:def 7;

                          then (k + ((( indx (D2,D1,n1)) + 1) - 1)) <= ( indx (D2,D1,j)) by XREAL_1: 19;

                          then ((k + (( indx (D2,D1,n1)) + 1)) - 1) <= ( len LVf(D2)) by A442, XXREAL_0: 2;

                          then (k + k2) in ( Seg ( len LVf(D2))) by A519, A520, FINSEQ_1: 1;

                          then

                           A521: (k + k2) in ( Seg ( len D2)) by INTEGRA1:def 7;

                          then (k + k2) in ( dom D2) by FINSEQ_1:def 3;

                          then

                           A522: (( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . k) = (( lower_bound ( rng (f | ( divset (D2,(k + k2)))))) * ( vol ( divset (D2,(k + k2))))) by A518, A520, INTEGRA1:def 7;

                          

                           A523: k in ( Seg ( len MD2)) by A517, INTEGRA1:def 7;

                          

                           A524: ( lower_bound ( divset (MD2,k))) = ( lower_bound ( divset (D2,(k + k2)))) & ( upper_bound ( divset (MD2,k))) = ( upper_bound ( divset (D2,(k + k2))))

                          proof

                            (k + k2) >= (1 + 1) by A260, A515, A520, XREAL_1: 7;

                            then

                             A525: (k + k2) > 1 by NAT_1: 13;

                            

                             A526: k in ( dom MD2) by A523, FINSEQ_1:def 3;

                            

                             A527: (k + k2) in ( dom D2) by A521, FINSEQ_1:def 3;

                            per cases ;

                              suppose

                               A528: k = 1;

                              then ( upper_bound ( divset (MD2,k))) = (MD2 . k) by A526, INTEGRA1:def 4;

                              then

                               A529: ( upper_bound ( divset (MD2,k))) = (D2 . ((k + (( indx (D2,D1,n1)) + 1)) - 1)) by A272, A263, A358, A388, A390, A445, A515, A516, FINSEQ_6: 122;

                              

                               A530: ( lower_bound ( divset (D2,(k + k2)))) = (D2 . ((k + k2) - 1)) by A525, A527, INTEGRA1:def 4;

                              ( lower_bound ( divset (MD2,k))) = (D1 . n1) by A354, A526, A528, INTEGRA1:def 4;

                              hence thesis by A13, A241, A520, A525, A527, A528, A529, A530, INTEGRA1:def 4, INTEGRA1:def 19;

                            end;

                              suppose

                               A531: k <> 1;

                              then ( upper_bound ( divset (MD2,k))) = (MD2 . k) by A526, INTEGRA1:def 4;

                              then

                               A532: ( upper_bound ( divset (MD2,k))) = (D2 . ((k + (( indx (D2,D1,n1)) + 1)) - 1)) by A272, A263, A358, A388, A390, A445, A515, A516, FINSEQ_6: 122;

                              

                               A533: (k - 1) <= ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A445, A516, XREAL_1: 146, XXREAL_0: 2;

                              

                               A534: ( lower_bound ( divset (MD2,k))) = (MD2 . (k - 1)) by A526, A531, INTEGRA1:def 4;

                              

                               A535: (k - 1) in ( dom MD2) by A526, A531, INTEGRA1: 7;

                              then 1 <= (k - 1) by FINSEQ_3: 25;

                              then ( lower_bound ( divset (MD2,k))) = (D2 . (((k - 1) + (( indx (D2,D1,n1)) + 1)) - 1)) by A272, A263, A358, A388, A390, A535, A533, A534, FINSEQ_6: 122;

                              hence thesis by A520, A525, A527, A532, INTEGRA1:def 4;

                            end;

                          end;

                          ( divset (MD2,k)) = [.( lower_bound ( divset (MD2,k))), ( upper_bound ( divset (MD2,k))).] by INTEGRA1: 4;

                          then

                           A536: ( divset (MD2,k)) = ( divset (D2,(k + k2))) by A524, INTEGRA1: 4;

                          k in ( dom MD2) by A523, FINSEQ_1:def 3;

                          then ( divset (D2,(k + k2))) c= B by A536, INTEGRA1: 8;

                          then

                           A537: ( rng (f | ( divset (D2,(k + k2))))) = ( rng (g | ( divset (D2,(k + k2))))) by FUNCT_1: 51;

                          k in ( dom MD2) by A523, FINSEQ_1:def 3;

                          hence thesis by A522, A536, A537, INTEGRA1:def 7;

                        end;

                        ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A237, INTEGRA2: 1;

                        then

                         A538: (D . (i + 1)) in ( divset (MD1,( len MD1))) by A374, A512, INTEGRA2: 1;

                        (j - 1) in ( dom D1) by A236, A245, A280, INTEGRA1: 7;

                        then (D1 . n1) < (D1 . (j - 1)) by A241, A441, SEQM_3:def 1;

                        then (D . (i + 1)) > ( lower_bound B) by A354, A511, XXREAL_0: 2;

                        then (( Sum ( lower_volume (g,MD2))) - ( Sum ( lower_volume (g,MD1)))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta MD1)) by A356, A431, A488, A538, A387, Th13;

                        then

                         A539: (( Sum ( lower_volume (g,MD2))) - ( Sum ( lower_volume (g,MD1)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) by A440, XXREAL_0: 2;

                        (( indx (D2,D1,n1)) + 1) <= ( len LVf(D2)) by A386, INTEGRA1:def 7;

                        then ( len ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) = ((( indx (D2,D1,j)) -' (( indx (D2,D1,n1)) + 1)) + 1) by A262, A388, A442, A513, FINSEQ_6: 118;

                        then ( len ( lower_volume (g,MD2))) = ( len ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) by A272, A385, A445, XREAL_1: 233, XXREAL_0: 2;

                        then

                         A540: ( Sum ( lower_volume (g,MD2))) = ( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) by A514, FINSEQ_1: 14;

                        (n1 + 1) <= ( len LVf(D1)) by A267, INTEGRA1:def 7;

                        

                        then ( len ( mid ( LVf(D1),(n1 + 1),j))) = ((j -' (n1 + 1)) + 1) by A257, A265, A244, A407, FINSEQ_6: 118

                        .= ((j - (n1 + 1)) + 1) by A257, XREAL_1: 233;

                        then ( Sum ( lower_volume (g,MD1))) = ( Sum ( mid ( LVf(D1),(n1 + 1),j))) by A371, A408, FINSEQ_1: 14;

                        hence thesis by A539, A510, A540, XXREAL_0: 2;

                      end;

                    end;

                    hence thesis;

                  end;

                  then

                   A541: (( PLf(D2,indx) - PLf(D1,n1)) + (( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) - ( Sum ( mid ( LVf(D1),(n1 + 1),j))))) <= (((i * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) + ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))) by A243, XREAL_1: 7;

                  n1 < (n1 + 1) by NAT_1: 13;

                  then (D1 . n1) < (D1 . (n1 + 1)) by A241, A268, SEQM_3:def 1;

                  then ( indx (D2,D1,n1)) < ( indx (D2,D1,(n1 + 1))) by A259, A274, A269, A271, SEQ_4: 137;

                  then

                   A542: ( indx (D2,D1,n1)) < ( indx (D2,D1,j)) by A272, XXREAL_0: 2;

                  ( indx (D2,D1,n1)) in ( Seg ( len D2)) by A259, FINSEQ_1:def 3;

                  then ( indx (D2,D1,n1)) in ( Seg ( len LVf(D2))) by INTEGRA1:def 7;

                  then ( indx (D2,D1,n1)) in ( dom LVf(D2)) by FINSEQ_1:def 3;

                  

                  then PLf(D2,indx) = ( Sum ( LVf(D2) | ( indx (D2,D1,n1)))) by INTEGRA1:def 20

                  .= ( Sum ( mid ( LVf(D2),1,( indx (D2,D1,n1))))) by A260, FINSEQ_6: 116;

                  

                  then ( PLf(D2,indx) + ( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) = ( Sum (( mid ( LVf(D2),1,( indx (D2,D1,n1)))) ^ ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) by RVSUM_1: 75

                  .= ( Sum ( mid ( LVf(D2),1,( indx (D2,D1,j))))) by A260, A542, A264, INTEGRA2: 4

                  .= ( Sum ( LVf(D2) | ( indx (D2,D1,j)))) by A262, FINSEQ_6: 116;

                  then PLf(D2,indx) = ( PLf(D2,indx) + ( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) by A279, INTEGRA1:def 20;

                  then (( PLf(D2,indx) - PLf(D1,n1)) + (( Sum ( mid ( LVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) - ( Sum ( mid ( LVf(D1),(n1 + 1),j))))) = ( PLf(D2,indx) - PLf(D1,j)) by A278;

                  hence thesis by A236, A237, A541;

                end;

                hence thesis;

              end;

              for k be non zero Nat holds P[k] from NAT_1:sch 10( A40, A231);

              then P[i];

              hence thesis by A17;

            end;

            

             A543: ( len D1) in ( dom D1) by FINSEQ_5: 6;

            then (D1 . ( len D1)) = (D2 . ( indx (D2,D1,( len D1)))) by A13, INTEGRA1:def 19;

            then ( upper_bound A) = (D2 . ( indx (D2,D1,( len D1)))) by INTEGRA1:def 2;

            then

             A544: (D2 . ( len D2)) = (D2 . ( indx (D2,D1,( len D1)))) by INTEGRA1:def 2;

            ( len D) in ( dom D) by FINSEQ_5: 6;

            then

            consider j such that

             A545: j in ( dom D1) and

             A546: (D . ( len D)) in ( divset (D1,j)) and

             A547: ( PLf(D2,indx) - PLf(D1,j)) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A16;

            

             A548: j = ( len D1)

            proof

              assume

               A549: j <> ( len D1);

              j <= ( len D1) by A545, FINSEQ_3: 25;

              then j < ( len D1) by A549, XXREAL_0: 1;

              then (D1 . j) < (D1 . ( len D1)) by A545, A543, SEQM_3:def 1;

              then

               A550: (D1 . j) < ( upper_bound A) by INTEGRA1:def 2;

              

               A551: ( upper_bound ( divset (D1,j))) < ( upper_bound A)

              proof

                per cases ;

                  suppose j = 1;

                  hence thesis by A545, A550, INTEGRA1:def 4;

                end;

                  suppose j <> 1;

                  hence thesis by A545, A550, INTEGRA1:def 4;

                end;

              end;

              (D . ( len D)) <= ( upper_bound ( divset (D1,j))) by A546, INTEGRA2: 1;

              hence contradiction by A551, INTEGRA1:def 2;

            end;

            ( indx (D2,D1,( len D1))) in ( dom D2) by A13, A543, INTEGRA1:def 19;

            then ( indx (D2,D1,( len D1))) = ( len D2) by A15, A544, SEQ_4: 138;

            then ( PLf(D2,len) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A547, A548, INTEGRA1: 43;

            hence thesis by INTEGRA1: 43;

          end;

          hence thesis by A12, A13, A14;

        end;

        hence thesis;

      end;

      hence thesis;

    end;

    theorem :: INTEGRA3:23

    

     Th22: (f | A) is bounded implies for D, D1 holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & 0 <= (( upper_sum (f,D)) - ( upper_sum (f,D2))) & 0 <= (( upper_sum (f,D1)) - ( upper_sum (f,D2)))

    proof

      assume

       A1: (f | A) is bounded;

      let D, D1;

      consider D2 such that

       A3: D <= D2 and

       A4: D1 <= D2 and

       A5: ( rng D2) = (( rng D1) \/ ( rng D)) by Th4;

      

       A6: (( upper_sum (f,D1)) - ( upper_sum (f,D2))) >= 0 by A1, A4, INTEGRA1: 45, XREAL_1: 48;

      (( upper_sum (f,D)) - ( upper_sum (f,D2))) >= 0 by A1, A3, INTEGRA1: 45, XREAL_1: 48;

      hence thesis by A3, A4, A5, A6;

    end;

    theorem :: INTEGRA3:24

    

     Th23: (f | A) is bounded implies for D, D1 st ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D)))) holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & (( upper_sum (f,D1)) - ( upper_sum (f,D2))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

    proof

      assume

       A1: (f | A) is bounded;

      then

       A2: for D, D1 holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & 0 <= (( upper_sum (f,D)) - ( upper_sum (f,D2))) & 0 <= (( upper_sum (f,D1)) - ( upper_sum (f,D2))) by Th22;

      let D, D1;

      assume

       A8: ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D))));

      ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & (( upper_sum (f,D1)) - ( upper_sum (f,D2))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

      proof

        consider D2 be Division of A such that

         A9: D <= D2 and

         A10: D1 <= D2 and

         A11: ( rng D2) = (( rng D1) \/ ( rng D)) and 0 <= (( upper_sum (f,D)) - ( upper_sum (f,D2))) and 0 <= (( upper_sum (f,D1)) - ( upper_sum (f,D2))) by A2;

        (( upper_sum (f,D1)) - ( upper_sum (f,D2))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

        proof

          deffunc UVf( Division of A) = ( upper_volume (f,$1));

          deffunc PUf( Division of A, Nat) = (( PartSums ( upper_volume (f,$1))) . $2);

          

           A12: ( len D2) in ( dom D2) by FINSEQ_5: 6;

          

           A13: for i st i in ( dom D) holds ex j st j in ( dom D1) & (D . i) in ( divset (D1,j)) & ( PUf(D1,j) - PUf(D2,indx)) <= ((i * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

          proof

            defpred P[ non zero Nat] means $1 in ( dom D) implies ex j st j in ( dom D1) & (D . $1) in ( divset (D1,j)) & ( PUf(D1,j) - PUf(D2,indx)) <= (($1 * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1));

            let i;

            assume

             A14: i in ( dom D);

            then

             A15: i in ( Seg ( len D)) by FINSEQ_1:def 3;

            

             A16: for i, j st i in ( dom D) & j in ( dom D1) & (D . i) in ( divset (D1,j)) holds j >= 2

            proof

              let i, j;

              assume

               A17: i in ( dom D);

              assume that

               A18: j in ( dom D1) and

               A19: (D . i) in ( divset (D1,j));

              assume j < 2;

              then j < (1 + 1);

              then

               A20: j <= 1 by NAT_1: 13;

              j in ( Seg ( len D1)) by A18, FINSEQ_1:def 3;

              then j >= 1 by FINSEQ_1: 1;

              then j = 1 by A20, XXREAL_0: 1;

              then

               A21: ( lower_bound ( divset (D1,j))) = ( lower_bound A) by A18, INTEGRA1:def 4;

              

               A22: (D . i) <= ( upper_bound ( divset (D1,j))) by A19, INTEGRA2: 1;

              ( delta D1) >= ( min ( rng ( upper_volume (( chi (A,A)),D))))

              proof

                per cases ;

                  suppose

                   A23: i = 1;

                  ( len D) in ( Seg ( len D)) by FINSEQ_1: 3;

                  then 1 <= ( len D) by FINSEQ_1: 1;

                  then

                   A24: 1 in ( dom D) by FINSEQ_3: 25;

                  then

                   A25: ( lower_bound ( divset (D,1))) = ( lower_bound A) by INTEGRA1:def 4;

                  1 in ( Seg ( len D)) by A24, FINSEQ_1:def 3;

                  then 1 in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by INTEGRA1:def 6;

                  then

                   A26: 1 in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

                  ( vol ( divset (D,1))) = (( upper_volume (( chi (A,A)),D)) . 1) by A24, INTEGRA1: 20;

                  then ( vol ( divset (D,1))) in ( rng ( upper_volume (( chi (A,A)),D))) by A26, FUNCT_1:def 3;

                  then

                   A27: ( vol ( divset (D,1))) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by XXREAL_2:def 7;

                  

                   A28: ( upper_bound ( divset (D,1))) = (D . 1) by A24, INTEGRA1:def 4;

                  (( upper_bound ( divset (D1,j))) - ( lower_bound A)) >= ((D . 1) - ( lower_bound A)) by A22, A23, XREAL_1: 9;

                  then ( vol ( divset (D1,j))) >= (( upper_bound ( divset (D,1))) - ( lower_bound ( divset (D,1)))) by A21, A25, A28, INTEGRA1:def 5;

                  then

                   A29: ( vol ( divset (D1,j))) >= ( vol ( divset (D,1))) by INTEGRA1:def 5;

                  ( vol ( divset (D1,j))) <= ( delta D1) by A18, Lm5;

                  then ( delta D1) >= ( vol ( divset (D,1))) by A29, XXREAL_0: 2;

                  hence thesis by A27, XXREAL_0: 2;

                end;

                  suppose

                   A30: i <> 1;

                  then (D . (i - 1)) in A by A17, INTEGRA1: 7;

                  then

                   A31: ( lower_bound A) <= (D . (i - 1)) by INTEGRA2: 1;

                  ( lower_bound ( divset (D,i))) = (D . (i - 1)) by A17, A30, INTEGRA1:def 4;

                  then

                   A32: (( upper_bound ( divset (D,i))) - ( lower_bound A)) >= (( upper_bound ( divset (D,i))) - ( lower_bound ( divset (D,i)))) by A31, XREAL_1: 10;

                  ( upper_bound ( divset (D,i))) = (D . i) by A17, A30, INTEGRA1:def 4;

                  then (( upper_bound ( divset (D1,j))) - ( lower_bound ( divset (D1,j)))) >= (( upper_bound ( divset (D,i))) - ( lower_bound A)) by A22, A21, XREAL_1: 9;

                  then (( upper_bound ( divset (D1,j))) - ( lower_bound ( divset (D1,j)))) >= (( upper_bound ( divset (D,i))) - ( lower_bound ( divset (D,i)))) by A32, XXREAL_0: 2;

                  then ( vol ( divset (D1,j))) >= (( upper_bound ( divset (D,i))) - ( lower_bound ( divset (D,i)))) by INTEGRA1:def 5;

                  then

                   A33: ( vol ( divset (D1,j))) >= ( vol ( divset (D,i))) by INTEGRA1:def 5;

                  i in ( Seg ( len D)) by A17, FINSEQ_1:def 3;

                  then i in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by INTEGRA1:def 6;

                  then

                   A34: i in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

                  ( vol ( divset (D,i))) = (( upper_volume (( chi (A,A)),D)) . i) by A17, INTEGRA1: 20;

                  then ( vol ( divset (D,i))) in ( rng ( upper_volume (( chi (A,A)),D))) by A34, FUNCT_1:def 3;

                  then

                   A35: ( vol ( divset (D,i))) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by XXREAL_2:def 7;

                  ( vol ( divset (D1,j))) <= ( delta D1) by A18, Lm5;

                  then ( delta D1) >= ( vol ( divset (D,i))) by A33, XXREAL_0: 2;

                  hence thesis by A35, XXREAL_0: 2;

                end;

              end;

              hence contradiction by A8;

            end;

            

             A36: P[1]

            proof

              ( len D) in ( Seg ( len D)) by FINSEQ_1: 3;

              then 1 <= ( len D) by FINSEQ_1: 1;

              then

               A37: 1 in ( dom D) by FINSEQ_3: 25;

              then

              consider j such that

               A38: j in ( dom D1) and

               A39: (D . 1) in ( divset (D1,j)) by Th3, INTEGRA1: 6;

              ( PUf(D1,j) - PUf(D2,indx)) <= ((1 * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1))

              proof

                

                 A40: j <> 1 by A16, A37, A38, A39;

                then

                reconsider j1 = (j - 1) as Element of NAT by A38, INTEGRA1: 7;

                

                 A41: j1 in ( dom D1) by A38, A40, INTEGRA1: 7;

                then j1 in ( Seg ( len D1)) by FINSEQ_1:def 3;

                then j1 in ( Seg ( len ( upper_volume (f,D1)))) by INTEGRA1:def 6;

                then

                 A42: j1 in ( dom ( upper_volume (f,D1))) by FINSEQ_1:def 3;

                

                 A43: (j - 1) in ( dom D1) by A38, A40, INTEGRA1: 7;

                then

                 A44: ( indx (D2,D1,j1)) in ( dom D2) by A10, INTEGRA1:def 19;

                then

                 A45: ( indx (D2,D1,j1)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                then

                 A46: 1 <= ( indx (D2,D1,j1)) by FINSEQ_1: 1;

                then ( mid (D2,1,( indx (D2,D1,j1)))) is increasing by A44, INTEGRA1: 35;

                then

                 A47: (D2 | ( indx (D2,D1,j1))) is increasing by A46, FINSEQ_6: 116;

                j < (j + 1) by NAT_1: 13;

                then j1 < j by XREAL_1: 19;

                then

                 A48: ( indx (D2,D1,j1)) < ( indx (D2,D1,j)) by A10, A38, A41, Th8;

                then

                 A49: (( indx (D2,D1,j1)) + 1) <= ( indx (D2,D1,j)) by NAT_1: 13;

                

                 A50: (( Sum ( mid (( upper_volume (f,D1)),j,j))) - ( Sum ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))

                proof

                  

                   A51: (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) <= 2

                  proof

                    reconsider ID1 = (( indx (D2,D1,j1)) + 1) as Element of NAT ;

                    reconsider ID2 = (ID1 + 1) as Element of NAT ;

                    assume (( indx (D2,D1,j)) - ( indx (D2,D1,j1))) > 2;

                    then

                     A52: (( indx (D2,D1,j1)) + (1 + 1)) < ( indx (D2,D1,j)) by XREAL_1: 20;

                    

                     A53: ID1 < ID2 by NAT_1: 13;

                    then ( indx (D2,D1,j1)) <= ID2 by NAT_1: 13;

                    then

                     A54: 1 <= ID2 by A46, XXREAL_0: 2;

                    

                     A55: ( indx (D2,D1,j)) in ( dom D2) by A10, A38, INTEGRA1:def 19;

                    then

                     A56: ( indx (D2,D1,j)) <= ( len D2) by FINSEQ_3: 25;

                    then ID2 <= ( len D2) by A52, XXREAL_0: 2;

                    then

                     A57: ID2 in ( dom D2) by A54, FINSEQ_3: 25;

                    then

                     A58: (D2 . ID2) < (D2 . ( indx (D2,D1,j))) by A52, A55, SEQM_3:def 1;

                    

                     A59: 1 <= ID1 by A46, NAT_1: 13;

                    

                     A60: (D1 . j) = (D2 . ( indx (D2,D1,j))) by A10, A38, INTEGRA1:def 19;

                    ID1 <= ( indx (D2,D1,j)) by A52, A53, XXREAL_0: 2;

                    then ID1 <= ( len D2) by A56, XXREAL_0: 2;

                    then

                     A61: ID1 in ( dom D2) by A59, FINSEQ_3: 25;

                    then

                     A62: (D2 . ID1) < (D2 . ID2) by A53, A57, SEQM_3:def 1;

                    ( indx (D2,D1,j1)) < ID1 by NAT_1: 13;

                    then

                     A63: (D2 . ( indx (D2,D1,j1))) < (D2 . ID1) by A44, A61, SEQM_3:def 1;

                    

                     A64: (D1 . j1) = (D2 . ( indx (D2,D1,j1))) by A10, A41, INTEGRA1:def 19;

                    

                     A65: not (D2 . ID1) in ( rng D1) & not (D2 . ID2) in ( rng D1)

                    proof

                      assume

                       A66: (D2 . ID1) in ( rng D1) or (D2 . ID2) in ( rng D1);

                      per cases by A66;

                        suppose (D2 . ID1) in ( rng D1);

                        then

                        consider n such that

                         A67: n in ( dom D1) and

                         A68: (D1 . n) = (D2 . ID1) by PARTFUN1: 3;

                        j1 < n by A41, A63, A64, A67, A68, SEQ_4: 137;

                        then

                         A69: j < (n + 1) by XREAL_1: 19;

                        (D2 . ID1) < (D2 . ( indx (D2,D1,j))) by A62, A58, XXREAL_0: 2;

                        then n < j by A38, A60, A67, A68, SEQ_4: 137;

                        hence contradiction by A69, NAT_1: 13;

                      end;

                        suppose (D2 . ID2) in ( rng D1);

                        then

                        consider n such that

                         A70: n in ( dom D1) and

                         A71: (D1 . n) = (D2 . ID2) by PARTFUN1: 3;

                        (D2 . ( indx (D2,D1,j1))) < (D2 . ID2) by A63, A62, XXREAL_0: 2;

                        then j1 < n by A41, A64, A70, A71, SEQ_4: 137;

                        then

                         A72: j < (n + 1) by XREAL_1: 19;

                        n < j by A38, A58, A60, A70, A71, SEQ_4: 137;

                        hence contradiction by A72, NAT_1: 13;

                      end;

                    end;

                    ( upper_bound ( divset (D1,j))) = (D1 . j) by A38, A40, INTEGRA1:def 4;

                    then

                     A73: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A10, A38, INTEGRA1:def 19;

                    ( lower_bound ( divset (D1,j))) = (D1 . j1) by A38, A40, INTEGRA1:def 4;

                    then

                     A74: ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A10, A41, INTEGRA1:def 19;

                    (D2 . ID2) in ( rng D2) by A57, FUNCT_1:def 3;

                    then

                     A75: (D2 . ID2) in ( rng D) by A11, A65, XBOOLE_0:def 3;

                    (D2 . ID1) in ( rng D2) by A61, FUNCT_1:def 3;

                    then

                     A76: (D2 . ID1) in ( rng D) by A11, A65, XBOOLE_0:def 3;

                    (D2 . ( indx (D2,D1,j1))) <= (D2 . ID2) by A63, A62, XXREAL_0: 2;

                    then (D2 . ID2) in ( divset (D1,j)) by A58, A74, A73, INTEGRA2: 1;

                    then

                     A77: (D2 . ID2) in (( rng D) /\ ( divset (D1,j))) by A75, XBOOLE_0:def 4;

                    (D2 . ID1) <= (D2 . ( indx (D2,D1,j))) by A62, A58, XXREAL_0: 2;

                    then (D2 . ID1) in ( divset (D1,j)) by A63, A74, A73, INTEGRA2: 1;

                    then (D2 . ID1) in (( rng D) /\ ( divset (D1,j))) by A76, XBOOLE_0:def 4;

                    hence contradiction by A8, A38, A53, A61, A57, A77, Th5, SEQ_4: 138;

                  end;

                  

                   A78: 1 <= (( indx (D2,D1,j1)) + 1) by A46, NAT_1: 13;

                  j <= ( len D1) by A38, FINSEQ_3: 25;

                  then

                   A79: j <= ( len ( upper_volume (f,D1))) by INTEGRA1:def 6;

                  

                   A80: 1 <= j by A38, FINSEQ_3: 25;

                  then

                   A81: (( mid (( upper_volume (f,D1)),j,j)) . 1) = (( upper_volume (f,D1)) . j) by A79, FINSEQ_6: 118;

                  reconsider uv = (( upper_volume (f,D1)) . j) as Element of REAL by XREAL_0:def 1;

                  ((j -' j) + 1) = 1 by Lm1;

                  then ( len ( mid (( upper_volume (f,D1)),j,j))) = 1 by A80, A79, FINSEQ_6: 118;

                  then ( mid (( upper_volume (f,D1)),j,j)) = <*uv*> by A81, FINSEQ_1: 40;

                  then

                   A82: ( Sum ( mid (( upper_volume (f,D1)),j,j))) = (( upper_volume (f,D1)) . j) by FINSOP_1: 11;

                  ( indx (D2,D1,j)) in ( dom D2) by A10, A38, INTEGRA1:def 19;

                  then

                   A83: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                  then

                   A84: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

                  ( indx (D2,D1,j)) in ( Seg ( len ( upper_volume (f,D2)))) by A83, INTEGRA1:def 6;

                  then

                   A85: ( indx (D2,D1,j)) <= ( len ( upper_volume (f,D2))) by FINSEQ_1: 1;

                  then

                   A86: (( indx (D2,D1,j1)) + 1) <= ( len ( upper_volume (f,D2))) by A49, XXREAL_0: 2;

                  then (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( upper_volume (f,D2)))) by A78, FINSEQ_1: 1;

                  then

                   A87: (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 6;

                  then

                   A88: (( indx (D2,D1,j1)) + 1) in ( dom D2) by FINSEQ_1:def 3;

                  (( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) = (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) by A49, XREAL_1: 233;

                  then ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) <= 2 by A51;

                  then

                   A89: ( len ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) <= 2 by A49, A84, A85, A78, A86, FINSEQ_6: 118;

                  ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) >= ( 0 + 1) by XREAL_1: 6;

                  then

                   A90: 1 <= ( len ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) by A49, A84, A85, A78, A86, FINSEQ_6: 118;

                  now

                    per cases by A90, A89, Lm2;

                      suppose

                       A91: ( len ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 1;

                      ( upper_bound ( divset (D1,j))) = (D1 . j) by A38, A40, INTEGRA1:def 4;

                      then

                       A92: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A10, A38, INTEGRA1:def 19;

                      ( lower_bound ( divset (D1,j))) = (D1 . j1) by A38, A40, INTEGRA1:def 4;

                      then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A10, A41, INTEGRA1:def 19;

                      then

                       A93: ( divset (D1,j)) = [.(D2 . ( indx (D2,D1,j1))), (D2 . ( indx (D2,D1,j))).] by A92, INTEGRA1: 4;

                      

                       A94: ( delta D1) >= 0 by Th9;

                      

                       A95: (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                      

                       A96: ( indx (D2,D1,j)) in ( dom D2) by A10, A38, INTEGRA1:def 19;

                      ( len ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) by A49, A84, A85, A78, A86, FINSEQ_6: 118;

                      then

                       A97: (( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) = 0 by A49, A91, XREAL_1: 233;

                      then ( indx (D2,D1,j)) <> 1 by A45, FINSEQ_1: 1;

                      then

                       A98: ( upper_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . ( indx (D2,D1,j))) by A96, INTEGRA1:def 4;

                      (( indx (D2,D1,j)) - 1) = ( indx (D2,D1,j1)) by A97;

                      then ( lower_bound ( divset (D2,( indx (D2,D1,j))))) = (D2 . ( indx (D2,D1,j1))) by A46, A97, A96, INTEGRA1:def 4;

                      then

                       A99: ( divset (D2,( indx (D2,D1,j)))) = ( divset (D1,j)) by A93, A98, INTEGRA1: 4;

                      reconsider uv = (( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) as Element of REAL by XREAL_0:def 1;

                      (( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) by A84, A85, A78, A86, FINSEQ_6: 118;

                      then ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*uv*> by A91, FINSEQ_1: 40;

                      

                      then ( Sum ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = (( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) by FINSOP_1: 11

                      .= (( upper_bound ( rng (f | ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))) by A88, INTEGRA1:def 6

                      .= ( Sum ( mid (( upper_volume (f,D1)),j,j))) by A38, A82, A97, A99, INTEGRA1:def 6;

                      hence thesis by A94, A95;

                    end;

                      suppose

                       A100: ( len ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = 2;

                      

                       A101: (( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 1) = (( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) by A84, A85, A78, A86, FINSEQ_6: 118;

                      

                       A102: (2 + (( indx (D2,D1,j1)) + 1)) >= ( 0 + 1) by XREAL_1: 7;

                      (( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) . 2) = ( UVf(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) -' 1)) by A49, A84, A85, A78, A86, A100, FINSEQ_6: 118

                      .= ( UVf(D2) . ((2 + (( indx (D2,D1,j1)) + 1)) - 1)) by A102, XREAL_1: 233

                      .= ( UVf(D2) . (( indx (D2,D1,j1)) + (1 + 1)));

                      then ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))) = <*(( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)), (( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 2))*> by A100, A101, FINSEQ_1: 44;

                      then

                       A103: ( Sum ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j))))) = ((( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 1)) + (( upper_volume (f,D2)) . (( indx (D2,D1,j1)) + 2))) by RVSUM_1: 77;

                      

                       A104: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) >= 0 by INTEGRA1: 9;

                      ( upper_bound ( divset (D1,j))) = (D1 . j) by A38, A40, INTEGRA1:def 4;

                      then

                       A105: ( upper_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j))) by A10, A38, INTEGRA1:def 19;

                      

                       A106: ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) >= 0 by INTEGRA1: 9;

                      ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A49, A84, A85, A78, A86, A100, FINSEQ_6: 118;

                      

                      then

                       A107: 2 = ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) by A49, XREAL_1: 233

                      .= (( indx (D2,D1,j)) - ( indx (D2,D1,j1)));

                      then

                       A108: (( indx (D2,D1,j1)) + 2) in ( dom D2) by A10, A38, INTEGRA1:def 19;

                      ( lower_bound ( divset (D1,j))) = (D1 . j1) by A38, A40, INTEGRA1:def 4;

                      then ( lower_bound ( divset (D1,j))) = (D2 . ( indx (D2,D1,j1))) by A10, A41, INTEGRA1:def 19;

                      then

                       A109: ( vol ( divset (D1,j))) = ((((D2 . (( indx (D2,D1,j1)) + 2)) - (D2 . (( indx (D2,D1,j1)) + 1))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A105, A107, INTEGRA1:def 5;

                      (( indx (D2,D1,j1)) + 1) in ( Seg ( len ( upper_volume (f,D2)))) by A78, A86, FINSEQ_1: 1;

                      then (( indx (D2,D1,j1)) + 1) in ( Seg ( len D2)) by INTEGRA1:def 6;

                      then

                       A110: (( indx (D2,D1,j1)) + 1) in ( dom D2) by FINSEQ_1:def 3;

                      

                       A111: (( indx (D2,D1,j1)) + 1) <> 1 by A46, NAT_1: 13;

                      then

                       A112: ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A110, INTEGRA1:def 4;

                      ((( indx (D2,D1,j1)) + 1) - 1) = (( indx (D2,D1,j1)) + 0 );

                      then

                       A113: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) = (D2 . ( indx (D2,D1,j1))) by A110, A111, INTEGRA1:def 4;

                      

                       A114: ((( indx (D2,D1,j1)) + 1) + 1) > 1 by A78, NAT_1: 13;

                      ((( indx (D2,D1,j1)) + 2) - 1) = (( indx (D2,D1,j1)) + 1);

                      then

                       A115: ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 1)) by A108, A114, INTEGRA1:def 4;

                      ( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 2)))) = (D2 . (( indx (D2,D1,j1)) + 2)) by A108, A114, INTEGRA1:def 4;

                      

                      then ( vol ( divset (D1,j))) = ((( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (D2 . (( indx (D2,D1,j1)) + 1))) - (D2 . ( indx (D2,D1,j1)))) by A115, A109, INTEGRA1:def 5

                      .= (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + (( upper_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))) - ( lower_bound ( divset (D2,(( indx (D2,D1,j1)) + 1)))))) by A113, A112;

                      then

                       A116: ( vol ( divset (D1,j))) = (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2))))) by INTEGRA1:def 5;

                      then

                       A117: (( upper_volume (f,D1)) . j) = (( upper_bound ( rng (f | ( divset (D1,j))))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 1)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))))) by A38, INTEGRA1:def 6;

                      

                       A118: (( Sum ( mid ( UVf(D1),j,j))) - ( Sum ( mid ( UVf(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * (( vol ( divset (D2,(( indx (D2,D1,j1)) + 2)))) + ( vol ( divset (D2,(( indx (D2,D1,j1)) + 1))))))

                      proof

                        set ID2 = (( indx (D2,D1,j1)) + 2);

                        set ID1 = (( indx (D2,D1,j1)) + 1);

                        set SR = ( upper_bound ( rng (f | ( divset (D2,ID1))))), VR = ( vol ( divset (D2,ID1)));

                        set B = ( vol ( divset (D2,ID1)));

                        set C = ( vol ( divset (D2,ID2)));

                        ( divset (D1,j)) c= A by A38, INTEGRA1: 8;

                        then

                         A119: ( upper_bound ( rng (f | ( divset (D1,j))))) <= ( upper_bound ( rng f)) by A1, Lm4;

                        ID1 in ( dom D2) by A87, FINSEQ_1:def 3;

                        then ( divset (D2,ID1)) c= A by INTEGRA1: 8;

                        then ( upper_bound ( rng (f | ( divset (D2,ID1))))) >= ( lower_bound ( rng f)) by A1, Lm4;

                        then

                         A120: (( upper_bound ( rng (f | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1)))) >= (( lower_bound ( rng f)) * ( vol ( divset (D2,ID1)))) by A104, XREAL_1: 64;

                        ((( indx (D2,D1,j)) -' (( indx (D2,D1,j1)) + 1)) + 1) = 2 by A49, A84, A85, A78, A86, A100, FINSEQ_6: 118;

                        

                        then

                         A121: 2 = ((( indx (D2,D1,j)) - (( indx (D2,D1,j1)) + 1)) + 1) by A49, XREAL_1: 233

                        .= (( indx (D2,D1,j)) - ( indx (D2,D1,j1)));

                        

                         A122: ( indx (D2,D1,j)) in ( dom D2) by A10, A38, INTEGRA1:def 19;

                        then ( divset (D2,ID2)) c= A by A121, INTEGRA1: 8;

                        then

                         A123: ( upper_bound ( rng (f | ( divset (D2,ID2))))) >= ( lower_bound ( rng f)) by A1, Lm4;

                        reconsider A = ( upper_bound ( rng (f | ( divset (D1,j))))) as Real;

                        

                         A124: ((( upper_volume (f,D1)) . j) - (A * B)) = (A * C) by A117;

                        (( upper_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID2)))) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,ID2)))) by A106, A119, XREAL_1: 64;

                        then ( Sum ( mid ( UVf(D1),j,j))) <= ((( upper_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) + (( upper_bound ( rng f)) * ( vol ( divset (D2,ID2))))) by A82, A124, XREAL_1: 20;

                        then

                         A125: (( Sum ( mid ( UVf(D1),j,j))) - (( upper_bound ( rng f)) * ( vol ( divset (D2,ID2))))) <= (( upper_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) by XREAL_1: 20;

                        (( upper_bound ( rng (f | ( divset (D1,j))))) * ( vol ( divset (D2,ID1)))) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1)))) by A104, A119, XREAL_1: 64;

                        then (( Sum ( mid ( UVf(D1),j,j))) - (( upper_bound ( rng f)) * ( vol ( divset (D2,ID2))))) <= (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1)))) by A125, XXREAL_0: 2;

                        then

                         A126: ( Sum ( mid ( UVf(D1),j,j))) <= ((( upper_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1))))) by XREAL_1: 20;

                        ( Sum ( mid ( UVf(D2),ID1,( indx (D2,D1,j))))) = ((( upper_bound ( rng (f | ( divset (D2,ID2))))) * ( vol ( divset (D2,ID2)))) + ( UVf(D2) . ID1)) by A103, A122, A121, INTEGRA1:def 6

                        .= ((( upper_bound ( rng (f | ( divset (D2,ID2))))) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng (f | ( divset (D2,ID1))))) * ( vol ( divset (D2,ID1))))) by A88, INTEGRA1:def 6;

                        then (( Sum ( mid ( UVf(D2),ID1,( indx (D2,D1,j))))) - (SR * VR)) >= (( lower_bound ( rng f)) * ( vol ( divset (D2,ID2)))) by A106, A123, XREAL_1: 64;

                        then ( Sum ( mid ( UVf(D2),ID1,( indx (D2,D1,j))))) >= ((( lower_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (SR * VR)) by XREAL_1: 19;

                        then (( Sum ( mid ( UVf(D2),ID1,( indx (D2,D1,j))))) - (( lower_bound ( rng f)) * ( vol ( divset (D2,ID2))))) >= (SR * VR) by XREAL_1: 19;

                        then (( Sum ( mid ( UVf(D2),ID1,( indx (D2,D1,j))))) - (( lower_bound ( rng f)) * ( vol ( divset (D2,ID2))))) >= (( lower_bound ( rng f)) * VR) by A120, XXREAL_0: 2;

                        then ( Sum ( mid ( UVf(D2),ID1,( indx (D2,D1,j))))) >= ((( lower_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng f)) * VR)) by XREAL_1: 19;

                        then (( Sum ( mid ( UVf(D1),j,j))) - ( Sum ( mid ( UVf(D2),ID1,( indx (D2,D1,j)))))) <= (((( upper_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( upper_bound ( rng f)) * ( vol ( divset (D2,ID1))))) - ((( lower_bound ( rng f)) * ( vol ( divset (D2,ID2)))) + (( lower_bound ( rng f)) * ( vol ( divset (D2,ID1)))))) by A126, XREAL_1: 13;

                        hence thesis;

                      end;

                      (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                      then ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( vol ( divset (D1,j)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1)) by A38, Lm5, XREAL_1: 64;

                      hence thesis by A116, A118, XXREAL_0: 2;

                    end;

                  end;

                  hence thesis;

                end;

                j < (j + 1) by NAT_1: 13;

                then

                 A127: j1 < j by XREAL_1: 19;

                ( indx (D2,D1,j)) in ( dom D2) by A10, A38, INTEGRA1:def 19;

                then

                 A128: ( indx (D2,D1,j)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                then

                 A129: 1 <= ( indx (D2,D1,j)) by FINSEQ_1: 1;

                

                 A130: ( indx (D2,D1,j1)) <= ( len D2) by A45, FINSEQ_1: 1;

                then

                 A131: ( len (D2 | ( indx (D2,D1,j1)))) = ( indx (D2,D1,j1)) by FINSEQ_1: 59;

                

                 A132: j1 in ( Seg ( len D1)) by A43, FINSEQ_1:def 3;

                then

                 A133: j1 <= ( len D1) by FINSEQ_1: 1;

                for x1 be object st x1 in ( rng (D1 | j1)) holds x1 in ( rng (D2 | ( indx (D2,D1,j1))))

                proof

                  let x1 be object;

                  assume x1 in ( rng (D1 | j1));

                  then

                  consider k such that

                   A134: k in ( dom (D1 | j1)) and

                   A135: x1 = ((D1 | j1) . k) by PARTFUN1: 3;

                  k in ( Seg ( len (D1 | j1))) by A134, FINSEQ_1:def 3;

                  then

                   A136: k in ( Seg j1) by A133, FINSEQ_1: 59;

                  then

                   A137: k in ( dom D1) by A41, RFINSEQ: 6;

                  k <= j1 by A136, FINSEQ_1: 1;

                  then (D1 . k) <= (D1 . j1) by A43, A137, SEQ_4: 137;

                  then (D2 . ( indx (D2,D1,k))) <= (D1 . j1) by A10, A137, INTEGRA1:def 19;

                  then

                   A138: (D2 . ( indx (D2,D1,k))) <= (D2 . ( indx (D2,D1,j1))) by A10, A43, INTEGRA1:def 19;

                  

                   A139: ((D1 | j1) . k) = (D1 . k) by A41, A136, RFINSEQ: 6;

                  (D1 . k) in ( rng D1) by A137, FUNCT_1:def 3;

                  then x1 in ( rng D2) by A11, A135, A139, XBOOLE_0:def 3;

                  then

                  consider n such that

                   A140: n in ( dom D2) and

                   A141: x1 = (D2 . n) by PARTFUN1: 3;

                  (D2 . ( indx (D2,D1,k))) = (D2 . n) by A10, A135, A139, A137, A141, INTEGRA1:def 19;

                  then

                   A142: n <= ( indx (D2,D1,j1)) by A44, A140, A138, SEQM_3:def 1;

                  1 <= n by A140, FINSEQ_3: 25;

                  then

                   A143: n in ( Seg ( indx (D2,D1,j1))) by A142, FINSEQ_1: 1;

                  then n in ( Seg ( len (D2 | ( indx (D2,D1,j1))))) by A130, FINSEQ_1: 59;

                  then

                   A144: n in ( dom (D2 | ( indx (D2,D1,j1)))) by FINSEQ_1:def 3;

                  (D2 . n) = ((D2 | ( indx (D2,D1,j1))) . n) by A44, A143, RFINSEQ: 6;

                  hence thesis by A141, A144, FUNCT_1:def 3;

                end;

                then

                 A145: ( rng (D1 | j1)) c= ( rng (D2 | ( indx (D2,D1,j1))));

                

                 A146: 1 <= j1 by A132, FINSEQ_1: 1;

                ( lower_bound ( divset (D1,j))) <= (D . 1) by A39, INTEGRA2: 1;

                then

                 A147: (D1 . j1) <= (D . 1) by A38, A40, INTEGRA1:def 4;

                for x1 be object st x1 in ( rng (D2 | ( indx (D2,D1,j1)))) holds x1 in ( rng (D1 | j1))

                proof

                  let x1 be object;

                  assume x1 in ( rng (D2 | ( indx (D2,D1,j1))));

                  then

                  consider k such that

                   A148: k in ( dom (D2 | ( indx (D2,D1,j1)))) and

                   A149: x1 = ((D2 | ( indx (D2,D1,j1))) . k) by PARTFUN1: 3;

                  k in ( Seg ( len (D2 | ( indx (D2,D1,j1))))) by A148, FINSEQ_1:def 3;

                  then

                   A150: k in ( Seg ( indx (D2,D1,j1))) by A130, FINSEQ_1: 59;

                  then

                   A151: k in ( dom D2) by A44, RFINSEQ: 6;

                  

                   A152: ( len (D1 | j1)) = j1 by A133, FINSEQ_1: 59;

                  k <= ( indx (D2,D1,j1)) by A150, FINSEQ_1: 1;

                  then (D2 . k) <= (D2 . ( indx (D2,D1,j1))) by A44, A151, SEQ_4: 137;

                  then

                   A153: (D2 . k) <= (D1 . j1) by A10, A43, INTEGRA1:def 19;

                  

                   A154: (D2 . k) in ( rng D1) implies (D2 . k) in ( rng (D1 | j1))

                  proof

                    assume (D2 . k) in ( rng D1);

                    then

                    consider m such that

                     A155: m in ( dom D1) and

                     A156: (D2 . k) = (D1 . m) by PARTFUN1: 3;

                    m in ( Seg ( len D1)) by A155, FINSEQ_1:def 3;

                    then

                     A157: 1 <= m by FINSEQ_1: 1;

                    

                     A158: m <= j1 by A41, A153, A155, A156, SEQM_3:def 1;

                    then m in ( Seg ( len (D1 | j1))) by A152, A157, FINSEQ_1: 1;

                    then

                     A159: m in ( dom (D1 | j1)) by FINSEQ_1:def 3;

                    m in ( Seg j1) by A157, A158, FINSEQ_1: 1;

                    then (D2 . k) = ((D1 | j1) . m) by A41, A156, RFINSEQ: 6;

                    hence thesis by A159, FUNCT_1:def 3;

                  end;

                  

                   A160: (D2 . k) in ( rng D) implies (D2 . k) = (D1 . j1)

                  proof

                    assume (D2 . k) in ( rng D);

                    then

                    consider n such that

                     A161: n in ( dom D) and

                     A162: (D2 . k) = (D . n) by PARTFUN1: 3;

                    1 <= n by A161, FINSEQ_3: 25;

                    then (D . 1) <= (D2 . k) by A37, A161, A162, SEQ_4: 137;

                    then (D1 . j1) <= (D2 . k) by A147, XXREAL_0: 2;

                    hence thesis by A153, XXREAL_0: 1;

                  end;

                  

                   A163: (D2 . k) in ( rng D) implies (D2 . k) in ( rng (D1 | j1))

                  proof

                    j1 in ( Seg ( len (D1 | j1))) by A146, A152, FINSEQ_1: 1;

                    then j1 in ( dom (D1 | j1)) by FINSEQ_1:def 3;

                    then

                     A164: ((D1 | j1) . j1) in ( rng (D1 | j1)) by FUNCT_1:def 3;

                    assume

                     A165: (D2 . k) in ( rng D);

                    j1 in ( Seg j1) by A146, FINSEQ_1: 1;

                    hence thesis by A41, A160, A165, A164, RFINSEQ: 6;

                  end;

                  (D2 . k) in ( rng D2) by A151, FUNCT_1:def 3;

                  hence thesis by A11, A44, A149, A150, A163, A154, RFINSEQ: 6, XBOOLE_0:def 3;

                end;

                then ( rng (D2 | ( indx (D2,D1,j1)))) c= ( rng (D1 | j1));

                then

                 A166: ( rng (D2 | ( indx (D2,D1,j1)))) = ( rng (D1 | j1)) by A145, XBOOLE_0:def 10;

                ( mid (D1,1,j1)) is increasing by A38, A40, A146, INTEGRA1: 7, INTEGRA1: 35;

                then

                 A167: (D1 | j1) is increasing by A146, FINSEQ_6: 116;

                then

                 A168: (D2 | ( indx (D2,D1,j1))) = (D1 | j1) by A47, A166, Th6;

                

                 A169: for k st 1 <= k & k <= j1 holds k = ( indx (D2,D1,k))

                proof

                  let k;

                  assume that

                   A170: 1 <= k and

                   A171: k <= j1;

                  assume

                   A172: k <> ( indx (D2,D1,k));

                  now

                    per cases by A172, XXREAL_0: 1;

                      suppose

                       A173: k > ( indx (D2,D1,k));

                      k <= ( len D1) by A133, A171, XXREAL_0: 2;

                      then

                       A174: k in ( dom D1) by A170, FINSEQ_3: 25;

                      then ( indx (D2,D1,k)) in ( dom D2) by A10, INTEGRA1:def 19;

                      then ( indx (D2,D1,k)) in ( Seg ( len D2)) by FINSEQ_1:def 3;

                      then

                       A175: 1 <= ( indx (D2,D1,k)) by FINSEQ_1: 1;

                      

                       A176: ( indx (D2,D1,k)) < j1 by A171, A173, XXREAL_0: 2;

                      then

                       A177: ( indx (D2,D1,k)) in ( Seg j1) by A175, FINSEQ_1: 1;

                      ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A10, A41, A171, A174, Th7;

                      then ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A175, FINSEQ_1: 1;

                      then

                       A178: ((D2 | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) = (D2 . ( indx (D2,D1,k))) by A44, RFINSEQ: 6;

                      ( indx (D2,D1,k)) <= ( len D1) by A133, A176, XXREAL_0: 2;

                      then ( indx (D2,D1,k)) in ( dom D1) by A175, FINSEQ_3: 25;

                      then

                       A179: (D1 . k) > (D1 . ( indx (D2,D1,k))) by A173, A174, SEQM_3:def 1;

                      (D1 . k) = (D2 . ( indx (D2,D1,k))) by A10, A174, INTEGRA1:def 19;

                      hence contradiction by A41, A168, A178, A179, A177, RFINSEQ: 6;

                    end;

                      suppose

                       A180: k < ( indx (D2,D1,k));

                      k <= ( len D1) by A133, A171, XXREAL_0: 2;

                      then

                       A181: k in ( dom D1) by A170, FINSEQ_3: 25;

                      then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by A10, A41, A171, Th7;

                      then

                       A182: k <= ( indx (D2,D1,j1)) by A180, XXREAL_0: 2;

                      then k <= ( len D2) by A130, XXREAL_0: 2;

                      then

                       A183: k in ( dom D2) by A170, FINSEQ_3: 25;

                      k in ( Seg j1) by A170, A171, FINSEQ_1: 1;

                      then

                       A184: (D1 . k) = ((D1 | j1) . k) by A43, RFINSEQ: 6;

                      ( indx (D2,D1,k)) in ( dom D2) by A10, A181, INTEGRA1:def 19;

                      then

                       A185: (D2 . k) < (D2 . ( indx (D2,D1,k))) by A180, A183, SEQM_3:def 1;

                      

                       A186: k in ( Seg ( indx (D2,D1,j1))) by A170, A182, FINSEQ_1: 1;

                      (D1 . k) = (D2 . ( indx (D2,D1,k))) by A10, A181, INTEGRA1:def 19;

                      hence contradiction by A44, A168, A184, A185, A186, RFINSEQ: 6;

                    end;

                  end;

                  hence contradiction;

                end;

                

                 A187: for k be Nat st 1 <= k & k <= ( len (( upper_volume (f,D1)) | j1)) holds ((( upper_volume (f,D1)) | j1) . k) = ((( upper_volume (f,D2)) | ( indx (D2,D1,j1))) . k)

                proof

                  ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A44, FINSEQ_1:def 3;

                  then ( indx (D2,D1,j1)) in ( Seg ( len ( upper_volume (f,D2)))) by INTEGRA1:def 6;

                  then

                   A188: ( indx (D2,D1,j1)) in ( dom ( upper_volume (f,D2))) by FINSEQ_1:def 3;

                  let k be Nat;

                  assume that

                   A189: 1 <= k and

                   A190: k <= ( len (( upper_volume (f,D1)) | j1));

                  

                   A191: ( len ( upper_volume (f,D1))) = ( len D1) by INTEGRA1:def 6;

                  then

                   A192: k <= j1 by A133, A190, FINSEQ_1: 59;

                  then

                   A193: k in ( Seg j1) by A189, FINSEQ_1: 1;

                  then ( indx (D2,D1,k)) in ( Seg j1) by A169, A189, A192;

                  then

                   A194: ( indx (D2,D1,k)) in ( Seg ( indx (D2,D1,j1))) by A146, A169;

                  then ( indx (D2,D1,k)) <= ( indx (D2,D1,j1)) by FINSEQ_1: 1;

                  then

                   A195: ( indx (D2,D1,k)) <= ( len D2) by A130, XXREAL_0: 2;

                  k <= ( len D1) by A133, A192, XXREAL_0: 2;

                  then

                   A196: k in ( Seg ( len D1)) by A189, FINSEQ_1: 1;

                  then

                   A197: k in ( dom D1) by FINSEQ_1:def 3;

                  then

                   A198: ( indx (D2,D1,k)) in ( dom D2) by A10, INTEGRA1:def 19;

                  

                   A199: (D1 . k) = (D2 . ( indx (D2,D1,k))) by A10, A197, INTEGRA1:def 19;

                  

                   A200: ( lower_bound ( divset (D1,k))) = ( lower_bound ( divset (D2,( indx (D2,D1,k))))) & ( upper_bound ( divset (D1,k))) = ( upper_bound ( divset (D2,( indx (D2,D1,k)))))

                  proof

                    per cases ;

                      suppose

                       A201: k = 1;

                      then

                       A202: ( upper_bound ( divset (D1,k))) = (D1 . k) by A197, INTEGRA1:def 4;

                      

                       A203: ( lower_bound ( divset (D1,k))) = ( lower_bound A) by A197, A201, INTEGRA1:def 4;

                      ( indx (D2,D1,k)) = 1 by A146, A169, A201;

                      hence thesis by A198, A199, A203, A202, INTEGRA1:def 4;

                    end;

                      suppose

                       A204: k <> 1;

                      then

                      reconsider k1 = (k - 1) as Element of NAT by A197, INTEGRA1: 7;

                      k <= (k + 1) by NAT_1: 11;

                      then k1 <= k by XREAL_1: 20;

                      then

                       A205: k1 <= j1 by A192, XXREAL_0: 2;

                      

                       A206: (k - 1) in ( dom D1) by A197, A204, INTEGRA1: 7;

                      then 1 <= k1 by FINSEQ_3: 25;

                      then k1 = ( indx (D2,D1,k1)) by A169, A205;

                      then

                       A207: (D2 . (( indx (D2,D1,k)) - 1)) = (D2 . ( indx (D2,D1,k1))) by A169, A189, A192, A193;

                      

                       A208: ( indx (D2,D1,k)) <> 1 by A169, A189, A192, A193, A204;

                      then

                       A209: ( lower_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . (( indx (D2,D1,k)) - 1)) by A198, INTEGRA1:def 4;

                      

                       A210: ( upper_bound ( divset (D2,( indx (D2,D1,k))))) = (D2 . ( indx (D2,D1,k))) by A198, A208, INTEGRA1:def 4;

                      

                       A211: ( upper_bound ( divset (D1,k))) = (D1 . k) by A197, A204, INTEGRA1:def 4;

                      ( lower_bound ( divset (D1,k))) = (D1 . (k - 1)) by A197, A204, INTEGRA1:def 4;

                      hence thesis by A10, A197, A211, A206, A209, A210, A207, INTEGRA1:def 19;

                    end;

                  end;

                  ( divset (D2,( indx (D2,D1,k)))) = [.( lower_bound ( divset (D2,( indx (D2,D1,k))))), ( upper_bound ( divset (D2,( indx (D2,D1,k))))).] by INTEGRA1: 4;

                  then

                   A212: ( divset (D1,k)) = ( divset (D2,( indx (D2,D1,k)))) by A200, INTEGRA1: 4;

                  

                   A213: k in ( dom D1) by A196, FINSEQ_1:def 3;

                  j1 in ( Seg ( len ( upper_volume (f,D1)))) by A41, A191, FINSEQ_1:def 3;

                  then j1 in ( dom ( upper_volume (f,D1))) by FINSEQ_1:def 3;

                  

                  then

                   A214: ((( upper_volume (f,D1)) | j1) . k) = (( upper_volume (f,D1)) . k) by A193, RFINSEQ: 6

                  .= (( upper_bound ( rng (f | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A213, A212, INTEGRA1:def 6;

                  1 <= ( indx (D2,D1,k)) by A169, A189, A192, A193;

                  then ( indx (D2,D1,k)) in ( Seg ( len D2)) by A195, FINSEQ_1: 1;

                  then

                   A215: ( indx (D2,D1,k)) in ( dom D2) by FINSEQ_1:def 3;

                  ((( upper_volume (f,D2)) | ( indx (D2,D1,j1))) . k) = ((( upper_volume (f,D2)) | ( indx (D2,D1,j1))) . ( indx (D2,D1,k))) by A169, A189, A192, A193

                  .= (( upper_volume (f,D2)) . ( indx (D2,D1,k))) by A194, A188, RFINSEQ: 6

                  .= (( upper_bound ( rng (f | ( divset (D2,( indx (D2,D1,k))))))) * ( vol ( divset (D2,( indx (D2,D1,k)))))) by A215, INTEGRA1:def 6;

                  hence thesis by A214;

                end;

                ( indx (D2,D1,j1)) in ( dom D2) by A10, A43, INTEGRA1:def 19;

                then ( indx (D2,D1,j1)) <= ( len D2) by FINSEQ_3: 25;

                then

                 A216: ( indx (D2,D1,j1)) <= ( len ( upper_volume (f,D2))) by INTEGRA1:def 6;

                j1 <= ( len D1) by A43, FINSEQ_3: 25;

                then

                 A217: j1 <= ( len ( upper_volume (f,D1))) by INTEGRA1:def 6;

                ( len (D2 | ( indx (D2,D1,j1)))) = ( len (D1 | j1)) by A47, A167, A166, Th6;

                then ( indx (D2,D1,j1)) = j1 by A133, A131, FINSEQ_1: 59;

                then ( len (( upper_volume (f,D1)) | j1)) = ( indx (D2,D1,j1)) by A217, FINSEQ_1: 59;

                then ( len (( upper_volume (f,D1)) | j1)) = ( len (( upper_volume (f,D2)) | ( indx (D2,D1,j1)))) by A216, FINSEQ_1: 59;

                then

                 A218: (( upper_volume (f,D2)) | ( indx (D2,D1,j1))) = (( upper_volume (f,D1)) | j1) by A187, FINSEQ_1: 14;

                

                 A219: j in ( Seg ( len D1)) by A38, FINSEQ_1:def 3;

                then

                 A220: 1 <= j by FINSEQ_1: 1;

                ( indx (D2,D1,j)) in ( Seg ( len UVf(D2))) by A128, INTEGRA1:def 6;

                then

                 A221: ( indx (D2,D1,j)) in ( dom UVf(D2)) by FINSEQ_1:def 3;

                ( indx (D2,D1,j)) <= ( len D2) by A128, FINSEQ_1: 1;

                then

                 A222: ( indx (D2,D1,j)) <= ( len UVf(D2)) by INTEGRA1:def 6;

                j in ( Seg ( len UVf(D1))) by A219, INTEGRA1:def 6;

                then

                 A223: j in ( dom UVf(D1)) by FINSEQ_1:def 3;

                j <= ( len D1) by A219, FINSEQ_1: 1;

                then

                 A224: j <= ( len UVf(D1)) by INTEGRA1:def 6;

                j1 in ( Seg ( len D1)) by A41, FINSEQ_1:def 3;

                then j1 in ( Seg ( len UVf(D1))) by INTEGRA1:def 6;

                then j1 in ( dom UVf(D1)) by FINSEQ_1:def 3;

                then PUf(D1,j1) = ( Sum ( UVf(D1) | j1)) by INTEGRA1:def 20;

                

                then ( PUf(D1,j1) + ( Sum ( mid ( UVf(D1),j,j)))) = ( Sum (( UVf(D1) | j1) ^ ( mid ( UVf(D1),j,j)))) by RVSUM_1: 75

                .= ( Sum (( mid ( UVf(D1),1,j1)) ^ ( mid ( UVf(D1),(j1 + 1),j)))) by A146, FINSEQ_6: 116

                .= ( Sum ( mid ( UVf(D1),1,j))) by A146, A224, A127, INTEGRA2: 4

                .= ( Sum ( UVf(D1) | j)) by A220, FINSEQ_6: 116;

                then

                 A225: ( PUf(D1,j1) + ( Sum ( mid (( upper_volume (f,D1)),j,j)))) = PUf(D1,j) by A223, INTEGRA1:def 20;

                ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A44, FINSEQ_1:def 3;

                then ( indx (D2,D1,j1)) in ( Seg ( len UVf(D2))) by INTEGRA1:def 6;

                then ( indx (D2,D1,j1)) in ( dom UVf(D2)) by FINSEQ_1:def 3;

                then PUf(D2,indx) = ( Sum ( UVf(D2) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20;

                

                then ( PUf(D2,indx) + ( Sum ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = ( Sum (( UVf(D2) | ( indx (D2,D1,j1))) ^ ( mid ( UVf(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by RVSUM_1: 75

                .= ( Sum (( mid ( UVf(D2),1,( indx (D2,D1,j1)))) ^ ( mid ( UVf(D2),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) by A46, FINSEQ_6: 116

                .= ( Sum ( mid ( UVf(D2),1,( indx (D2,D1,j))))) by A46, A48, A222, INTEGRA2: 4

                .= ( Sum ( UVf(D2) | ( indx (D2,D1,j)))) by A129, FINSEQ_6: 116;

                then

                 A226: ( PUf(D2,indx) + ( Sum ( mid (( upper_volume (f,D2)),(( indx (D2,D1,j1)) + 1),( indx (D2,D1,j)))))) = PUf(D2,indx) by A221, INTEGRA1:def 20;

                ( indx (D2,D1,j1)) in ( Seg ( len D2)) by A44, FINSEQ_1:def 3;

                then ( indx (D2,D1,j1)) in ( Seg ( len ( upper_volume (f,D2)))) by INTEGRA1:def 6;

                then ( indx (D2,D1,j1)) in ( dom ( upper_volume (f,D2))) by FINSEQ_1:def 3;

                

                then PUf(D2,indx) = ( Sum (( upper_volume (f,D2)) | ( indx (D2,D1,j1)))) by INTEGRA1:def 20

                .= PUf(D1,j1) by A218, A42, INTEGRA1:def 20;

                hence thesis by A50, A226, A225;

              end;

              hence thesis by A38, A39;

            end;

            reconsider i as non zero Element of NAT by A15, FINSEQ_1: 1;

            

             A227: for i be non zero Nat st P[i] holds P[(i + 1)]

            proof

              let i be non zero Nat;

              

               A228: i >= 1 by NAT_1: 14;

              assume

               A229: P[i];

               P[(i + 1)]

              proof

                

                 A230: i <= (i + 1) by NAT_1: 11;

                assume

                 A231: (i + 1) in ( dom D);

                then

                consider j such that

                 A232: j in ( dom D1) and

                 A233: (D . (i + 1)) in ( divset (D1,j)) by Th3, INTEGRA1: 6;

                

                 A234: (D2 . ( indx (D2,D1,j))) = (D1 . j) by A10, A232, INTEGRA1:def 19;

                (i + 1) in ( Seg ( len D)) by A231, FINSEQ_1:def 3;

                then (i + 1) <= ( len D) by FINSEQ_1: 1;

                then i <= ( len D) by A230, XXREAL_0: 2;

                then

                 A235: i in ( Seg ( len D)) by A228, FINSEQ_1: 1;

                then

                 A236: i in ( dom D) by FINSEQ_1:def 3;

                

                 A237: ( indx (D2,D1,j)) in ( dom D2) by A10, A232, INTEGRA1:def 19;

                then

                 A238: 1 <= ( indx (D2,D1,j)) by FINSEQ_3: 25;

                

                 A239: ( indx (D2,D1,j)) <= ( len D2) by A237, FINSEQ_3: 25;

                then

                 A240: ( indx (D2,D1,j)) <= ( len UVf(D2)) by INTEGRA1:def 6;

                consider n1 be Element of NAT such that

                 A241: n1 in ( dom D1) and

                 A242: (D . i) in ( divset (D1,n1)) and

                 A243: ( PUf(D1,n1) - PUf(D2,indx)) <= ((i * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A229, A235, FINSEQ_1:def 3;

                

                 A244: 1 <= (n1 + 1) by NAT_1: 12;

                

                 A245: n1 < j

                proof

                  assume

                   A246: n1 >= j;

                  now

                    per cases by A246, XXREAL_0: 1;

                      suppose

                       A247: n1 = j;

                      (D . i) in ( rng D) by A236, FUNCT_1:def 3;

                      then

                       A248: (D . i) in (( rng D) /\ ( divset (D1,j))) by A242, A247, XBOOLE_0:def 4;

                      (D . (i + 1)) in ( rng D) by A231, FUNCT_1:def 3;

                      then

                       A249: (D . (i + 1)) in (( rng D) /\ ( divset (D1,j))) by A233, XBOOLE_0:def 4;

                      (i + 1) > i by XREAL_1: 29;

                      hence contradiction by A8, A231, A232, A236, A248, A249, Th5, SEQ_4: 138;

                    end;

                      suppose n1 > j;

                      then

                       A250: n1 >= (j + 1) by NAT_1: 13;

                      then

                       A251: (n1 - 1) >= j by XREAL_1: 19;

                      1 <= j by A232, FINSEQ_3: 25;

                      then (1 + 1) <= (j + 1) by XREAL_1: 6;

                      then

                       A252: n1 <> 1 by A250, XXREAL_0: 2;

                      then (n1 - 1) in ( dom D1) by A241, INTEGRA1: 7;

                      then

                       A253: (D1 . j) <= (D1 . (n1 - 1)) by A232, A251, SEQ_4: 137;

                      

                       A254: ( upper_bound ( divset (D1,j))) = (D1 . j)

                      proof

                        per cases ;

                          suppose j = 1;

                          hence thesis by A232, INTEGRA1:def 4;

                        end;

                          suppose j <> 1;

                          hence thesis by A232, INTEGRA1:def 4;

                        end;

                      end;

                      

                       A255: ( lower_bound ( divset (D1,n1))) <= (D . i) by A242, INTEGRA2: 1;

                      ( lower_bound ( divset (D1,n1))) = (D1 . (n1 - 1)) by A241, A252, INTEGRA1:def 4;

                      then

                       A256: (D . i) >= (D1 . j) by A255, A253, XXREAL_0: 2;

                      

                       A257: i < (i + 1) by XREAL_1: 29;

                      (D . (i + 1)) <= ( upper_bound ( divset (D1,j))) by A233, INTEGRA2: 1;

                      then (D . i) >= (D . (i + 1)) by A254, A256, XXREAL_0: 2;

                      hence contradiction by A231, A236, A257, SEQM_3:def 1;

                    end;

                  end;

                  hence thesis;

                end;

                then

                 A258: (n1 + 1) <= j by NAT_1: 13;

                

                 A259: 1 <= n1 by A241, FINSEQ_3: 25;

                

                 A260: (D2 . ( indx (D2,D1,n1))) = (D1 . n1) by A10, A241, INTEGRA1:def 19;

                

                 A261: 1 <= j by A232, FINSEQ_3: 25;

                

                 A262: ( indx (D2,D1,n1)) in ( dom D2) by A10, A241, INTEGRA1:def 19;

                then

                 A263: 1 <= ( indx (D2,D1,n1)) by FINSEQ_3: 25;

                

                 A264: j <= ( len D1) by A232, FINSEQ_3: 25;

                then

                 A265: (n1 + 1) <= ( len D1) by A258, XXREAL_0: 2;

                then

                 A266: (n1 + 1) in ( dom D1) by A244, FINSEQ_3: 25;

                then

                 A267: (D2 . ( indx (D2,D1,(n1 + 1)))) = (D1 . (n1 + 1)) by A10, INTEGRA1:def 19;

                

                 A268: j <= ( len UVf(D1)) by A264, INTEGRA1:def 6;

                then j in ( Seg ( len UVf(D1))) by A261, FINSEQ_1: 1;

                then

                 A269: j in ( dom UVf(D1)) by FINSEQ_1:def 3;

                

                 A270: ( indx (D2,D1,(n1 + 1))) in ( dom D2) by A10, A266, INTEGRA1:def 19;

                then

                 A271: 1 <= ( indx (D2,D1,(n1 + 1))) by FINSEQ_3: 25;

                n1 in ( Seg ( len D1)) by A241, FINSEQ_1:def 3;

                then n1 in ( Seg ( len UVf(D1))) by INTEGRA1:def 6;

                then n1 in ( dom UVf(D1)) by FINSEQ_1:def 3;

                

                then PUf(D1,n1) = ( Sum ( UVf(D1) | n1)) by INTEGRA1:def 20

                .= ( Sum ( mid ( UVf(D1),1,n1))) by A259, FINSEQ_6: 116;

                

                then ( PUf(D1,n1) + ( Sum ( mid ( UVf(D1),(n1 + 1),j)))) = ( Sum (( mid ( UVf(D1),1,n1)) ^ ( mid ( UVf(D1),(n1 + 1),j)))) by RVSUM_1: 75

                .= ( Sum ( mid ( UVf(D1),1,j))) by A245, A259, A268, INTEGRA2: 4

                .= ( Sum ( UVf(D1) | j)) by A261, FINSEQ_6: 116;

                then

                 A272: PUf(D1,j) = ( PUf(D1,n1) + ( Sum ( mid ( UVf(D1),(n1 + 1),j)))) by A269, INTEGRA1:def 20;

                ( indx (D2,D1,j)) in ( Seg ( len D2)) by A237, FINSEQ_1:def 3;

                then ( indx (D2,D1,j)) in ( Seg ( len UVf(D2))) by INTEGRA1:def 6;

                then

                 A273: ( indx (D2,D1,j)) in ( dom UVf(D2)) by FINSEQ_1:def 3;

                

                 A274: ( indx (D2,D1,(n1 + 1))) <= ( len D2) by A270, FINSEQ_3: 25;

                (D1 . (n1 + 1)) <= (D1 . j) by A232, A258, A266, SEQ_4: 137;

                then

                 A275: ( indx (D2,D1,(n1 + 1))) <= ( indx (D2,D1,j)) by A270, A267, A237, A234, SEQM_3:def 1;

                then (1 + ( indx (D2,D1,(n1 + 1)))) <= (( indx (D2,D1,j)) + 1) by XREAL_1: 6;

                then 1 <= ((( indx (D2,D1,j)) + 1) - ( indx (D2,D1,(n1 + 1)))) by XREAL_1: 19;

                

                then

                 A276: (( mid (D2,( indx (D2,D1,(n1 + 1))),( indx (D2,D1,j)))) . 1) = (D2 . ((1 - 1) + ( indx (D2,D1,(n1 + 1))))) by A275, A271, A239, FINSEQ_6: 122

                .= (D1 . (n1 + 1)) by A10, A266, INTEGRA1:def 19;

                

                 A277: n1 >= 1 by A241, FINSEQ_3: 25;

                

                 A278: (j - n1) >= 1 by A258, XREAL_1: 19;

                (( Sum ( mid ( UVf(D1),(n1 + 1),j))) - ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))

                proof

                  per cases by A258, XXREAL_0: 1;

                    suppose

                     A279: (n1 + 1) = j;

                    

                     A280: (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) <= 2

                    proof

                      

                       A281: ( upper_bound ( divset (D1,j))) = (D1 . j) by A232, A245, A277, INTEGRA1:def 4;

                      

                       A282: ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A232, A245, A277, INTEGRA1:def 4;

                      

                       A283: 1 <= (( indx (D2,D1,n1)) + 1) by A263, NAT_1: 13;

                      assume (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) > 2;

                      then

                       A284: (( indx (D2,D1,n1)) + 2) < ( indx (D2,D1,j)) by XREAL_1: 20;

                      then

                       A285: (( indx (D2,D1,n1)) + 2) <= ( len D2) by A239, XXREAL_0: 2;

                      

                       A286: (( indx (D2,D1,n1)) + 1) < (( indx (D2,D1,n1)) + 2) by XREAL_1: 6;

                      then

                       A287: ( indx (D2,D1,n1)) < (( indx (D2,D1,n1)) + 2) by NAT_1: 13;

                      then 1 <= (( indx (D2,D1,n1)) + 2) by A263, XXREAL_0: 2;

                      then

                       A288: (( indx (D2,D1,n1)) + 2) in ( dom D2) by A285, FINSEQ_3: 25;

                      then

                       A289: (D2 . ( indx (D2,D1,j))) >= (D2 . (( indx (D2,D1,n1)) + 2)) by A237, A284, SEQ_4: 137;

                      

                       A290: not (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D1)

                      proof

                        assume (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D1);

                        then

                        consider k1 be Element of NAT such that

                         A291: k1 in ( dom D1) and

                         A292: (D2 . (( indx (D2,D1,n1)) + 2)) = (D1 . k1) by PARTFUN1: 3;

                        (D2 . (( indx (D2,D1,n1)) + 2)) < (D2 . ( indx (D2,D1,j))) by A237, A284, A288, SEQM_3:def 1;

                        then

                         A293: k1 < j by A232, A234, A291, A292, SEQ_4: 137;

                        (D2 . ( indx (D2,D1,n1))) < (D2 . (( indx (D2,D1,n1)) + 2)) by A262, A287, A288, SEQM_3:def 1;

                        then n1 < k1 by A241, A260, A291, A292, SEQ_4: 137;

                        hence contradiction by A279, A293, NAT_1: 13;

                      end;

                      (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D2) by A288, FUNCT_1:def 3;

                      then

                       A294: (D2 . (( indx (D2,D1,n1)) + 2)) in ( rng D) by A11, A290, XBOOLE_0:def 3;

                      

                       A295: ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A232, A245, A277, INTEGRA1:def 4;

                      

                       A296: ( upper_bound ( divset (D1,j))) = (D1 . j) by A232, A245, A277, INTEGRA1:def 4;

                      (D2 . (( indx (D2,D1,n1)) + 2)) >= (D2 . ( indx (D2,D1,n1))) by A262, A287, A288, SEQ_4: 137;

                      then (D2 . (( indx (D2,D1,n1)) + 2)) in ( divset (D1,j)) by A260, A234, A279, A295, A281, A289, INTEGRA2: 1;

                      then

                       A297: (D2 . (( indx (D2,D1,n1)) + 2)) in (( rng D) /\ ( divset (D1,j))) by A294, XBOOLE_0:def 4;

                      

                       A298: (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j)) by A284, A286, XXREAL_0: 2;

                      then (( indx (D2,D1,n1)) + 1) <= ( len D2) by A239, XXREAL_0: 2;

                      then

                       A299: (( indx (D2,D1,n1)) + 1) in ( dom D2) by A283, FINSEQ_3: 25;

                      then

                       A300: (D2 . ( indx (D2,D1,j))) >= (D2 . (( indx (D2,D1,n1)) + 1)) by A237, A298, SEQ_4: 137;

                      

                       A301: ( indx (D2,D1,n1)) < (( indx (D2,D1,n1)) + 1) by NAT_1: 13;

                      

                       A302: not (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D1)

                      proof

                        assume (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D1);

                        then

                        consider k1 be Element of NAT such that

                         A303: k1 in ( dom D1) and

                         A304: (D2 . (( indx (D2,D1,n1)) + 1)) = (D1 . k1) by PARTFUN1: 3;

                        (D2 . (( indx (D2,D1,n1)) + 1)) < (D2 . ( indx (D2,D1,j))) by A237, A298, A299, SEQM_3:def 1;

                        then

                         A305: k1 < j by A232, A234, A303, A304, SEQ_4: 137;

                        (D2 . ( indx (D2,D1,n1))) < (D2 . (( indx (D2,D1,n1)) + 1)) by A262, A301, A299, SEQM_3:def 1;

                        then n1 < k1 by A241, A260, A303, A304, SEQ_4: 137;

                        hence contradiction by A279, A305, NAT_1: 13;

                      end;

                      (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D2) by A299, FUNCT_1:def 3;

                      then

                       A306: (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D) by A11, A302, XBOOLE_0:def 3;

                      (D2 . (( indx (D2,D1,n1)) + 1)) >= (D2 . ( indx (D2,D1,n1))) by A262, A301, A299, SEQ_4: 137;

                      then (D2 . (( indx (D2,D1,n1)) + 1)) in ( divset (D1,j)) by A260, A234, A279, A282, A296, A300, INTEGRA2: 1;

                      then (D2 . (( indx (D2,D1,n1)) + 1)) in (( rng D) /\ ( divset (D1,j))) by A306, XBOOLE_0:def 4;

                      then (D2 . (( indx (D2,D1,n1)) + 1)) = (D2 . (( indx (D2,D1,n1)) + 2)) by A8, A232, A297, Th5;

                      hence contradiction by A286, A299, A288, SEQM_3:def 1;

                    end;

                    

                     A307: (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j)) implies (( indx (D2,D1,n1)) + 2) = ( indx (D2,D1,j))

                    proof

                      assume (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j));

                      then

                       A308: ((( indx (D2,D1,n1)) + 1) + 1) <= ( indx (D2,D1,j)) by NAT_1: 13;

                      (( indx (D2,D1,n1)) + 2) >= ( indx (D2,D1,j)) by A280, XREAL_1: 20;

                      hence thesis by A308, XXREAL_0: 1;

                    end;

                    

                     A309: 1 <= (( indx (D2,D1,n1)) + 1) by NAT_1: 12;

                    

                     A310: ( indx (D2,D1,j)) <= ( len UVf(D2)) by A239, INTEGRA1:def 6;

                    (D1 . n1) < (D1 . j) by A232, A241, A245, SEQM_3:def 1;

                    then

                     A311: ( indx (D2,D1,n1)) < ( indx (D2,D1,j)) by A262, A260, A237, A234, SEQ_4: 137;

                    then

                     A312: (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,j)) by NAT_1: 13;

                    then (( indx (D2,D1,n1)) + 1) <= ( len D2) by A239, XXREAL_0: 2;

                    then (( indx (D2,D1,n1)) + 1) <= ( len UVf(D2)) by INTEGRA1:def 6;

                    

                    then

                     A313: ( len ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) = ((( indx (D2,D1,j)) -' (( indx (D2,D1,n1)) + 1)) + 1) by A238, A312, A309, A310, FINSEQ_6: 118

                    .= ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A312, XREAL_1: 233

                    .= (( indx (D2,D1,j)) - ( indx (D2,D1,n1)));

                    (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,j)) by A311, NAT_1: 13;

                    then

                     A314: (( indx (D2,D1,n1)) + 1) = ( indx (D2,D1,j)) or (( indx (D2,D1,n1)) + 1) < ( indx (D2,D1,j)) by XXREAL_0: 1;

                    

                     A315: ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) >= (( lower_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1)))))

                    proof

                      now

                        per cases by A314, A307;

                          suppose

                           A316: (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) = 1;

                          (( indx (D2,D1,n1)) + 1) >= (1 + 1) by A263, XREAL_1: 6;

                          then

                           A317: (( indx (D2,D1,n1)) + 1) <> 1;

                          then ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . (( indx (D2,D1,n1)) + 1)) by A237, A316, INTEGRA1:def 4;

                          then

                           A318: ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D1 . j) by A10, A232, A316, INTEGRA1:def 19;

                          ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . ((( indx (D2,D1,n1)) + 1) - 1)) by A237, A316, A317, INTEGRA1:def 4;

                          then

                           A319: ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D1 . n1) by A10, A241, INTEGRA1:def 19;

                          ( lower_bound ( divset (D1,(n1 + 1)))) = (D1 . ((n1 + 1) - 1)) by A245, A277, A266, A279, INTEGRA1:def 4;

                          then

                           A320: ( divset (D2,(( indx (D2,D1,n1)) + 1))) = ( divset (D1,(n1 + 1))) by A245, A277, A266, A279, A319, A318, INTEGRA1:def 4;

                          

                           A321: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1)))) >= 0 by INTEGRA1: 9;

                          reconsider UV = ( UVf(D2) . (( indx (D2,D1,n1)) + 1)) as Element of REAL by XREAL_0:def 1;

                          1 = ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A316;

                          

                          then (( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . 1) = ( UVf(D2) . ((1 + (( indx (D2,D1,n1)) + 1)) - 1)) by A309, A310, FINSEQ_6: 122

                          .= ( UVf(D2) . (( indx (D2,D1,n1)) + 1));

                          then

                           A322: ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) = <*UV*> by A313, A316, FINSEQ_1: 40;

                          ( UVf(D2) . (( indx (D2,D1,n1)) + 1)) = (( upper_bound ( rng (f | ( divset (D2,(( indx (D2,D1,n1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) by A237, A316, INTEGRA1:def 6;

                          then ( UVf(D2) . (( indx (D2,D1,n1)) + 1)) >= (( lower_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1))))) by A1, A237, A316, A320, A321, Th19, XREAL_1: 64;

                          hence thesis by A322, FINSOP_1: 11;

                        end;

                          suppose

                           A323: (( indx (D2,D1,j)) - ( indx (D2,D1,n1))) = 2;

                          (( indx (D2,D1,n1)) + 2) >= (2 + 1) by A263, XREAL_1: 6;

                          then

                           A324: (( indx (D2,D1,n1)) + 2) <> 1;

                          then

                           A325: ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 2)))) = (D2 . ( indx (D2,D1,j))) by A237, A323, INTEGRA1:def 4;

                          ((( indx (D2,D1,n1)) + 2) - 1) = (( indx (D2,D1,n1)) + 1);

                          then ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 2)))) = (D2 . (( indx (D2,D1,n1)) + 1)) by A237, A323, A324, INTEGRA1:def 4;

                          then

                           A326: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2)))) = ((D1 . j) - (D2 . (( indx (D2,D1,n1)) + 1))) by A234, A325, INTEGRA1:def 5;

                          

                           A327: ( upper_bound ( divset (D1,(n1 + 1)))) = (D1 . (n1 + 1)) by A245, A277, A266, A279, INTEGRA1:def 4;

                          ( lower_bound ( divset (D1,(n1 + 1)))) = (D1 . ((n1 + 1) - 1)) by A245, A277, A266, A279, INTEGRA1:def 4;

                          then

                           A328: ( vol ( divset (D1,(n1 + 1)))) = ((D1 . (n1 + 1)) - (D1 . n1)) by A327, INTEGRA1:def 5;

                          

                           A329: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2)))) >= 0 by INTEGRA1: 9;

                          

                           A330: ( indx (D2,D1,j)) <= ( len UVf(D2)) by A239, INTEGRA1:def 6;

                          

                           A331: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1)))) >= 0 by INTEGRA1: 9;

                          

                           A332: 1 <= (( indx (D2,D1,n1)) + 1) by NAT_1: 12;

                          

                           A333: (( indx (D2,D1,n1)) + 1) <= (( indx (D2,D1,n1)) + 2) by XREAL_1: 6;

                          then (( indx (D2,D1,n1)) + 1) <= ( len D2) by A239, A323, XXREAL_0: 2;

                          then

                           A334: (( indx (D2,D1,n1)) + 1) in ( dom D2) by A332, FINSEQ_3: 25;

                          then ( UVf(D2) . (( indx (D2,D1,n1)) + 1)) = (( upper_bound ( rng (f | ( divset (D2,(( indx (D2,D1,n1)) + 1)))))) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) by INTEGRA1:def 6;

                          then

                           A335: ( UVf(D2) . (( indx (D2,D1,n1)) + 1)) >= (( lower_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) by A1, A334, A331, Th19, XREAL_1: 64;

                          ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) = (1 + 1) by A323;

                          

                          then

                           A336: (( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . 2) = ( UVf(D2) . ((2 + (( indx (D2,D1,n1)) + 1)) - 1)) by A332, A333, A330, FINSEQ_6: 122

                          .= ( UVf(D2) . ((( indx (D2,D1,n1)) + 0 ) + 2));

                          ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) >= 1 by A323;

                          

                          then (( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . 1) = ( UVf(D2) . ((1 + (( indx (D2,D1,n1)) + 1)) - 1)) by A323, A332, A333, A330, FINSEQ_6: 122

                          .= ( UVf(D2) . (( indx (D2,D1,n1)) + 1));

                          then ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) = <*( UVf(D2) . (( indx (D2,D1,n1)) + 1)), ( UVf(D2) . (( indx (D2,D1,n1)) + 2))*> by A313, A323, A336, FINSEQ_1: 44;

                          then

                           A337: ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) = (( UVf(D2) . (( indx (D2,D1,n1)) + 1)) + ( UVf(D2) . (( indx (D2,D1,n1)) + 2))) by RVSUM_1: 77;

                          

                           A338: (( indx (D2,D1,n1)) + 1) > 1 by A263, NAT_1: 13;

                          then

                           A339: ( upper_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . (( indx (D2,D1,n1)) + 1)) by A334, INTEGRA1:def 4;

                          ( lower_bound ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = (D2 . ((( indx (D2,D1,n1)) + 1) - 1)) by A334, A338, INTEGRA1:def 4;

                          then

                           A340: ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1)))) = ((D2 . (( indx (D2,D1,n1)) + 1)) - (D1 . n1)) by A260, A339, INTEGRA1:def 5;

                          ( UVf(D2) . (( indx (D2,D1,n1)) + 2)) = (( upper_bound ( rng (f | ( divset (D2,(( indx (D2,D1,n1)) + 2)))))) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2))))) by A237, A323, INTEGRA1:def 6;

                          then ( UVf(D2) . (( indx (D2,D1,n1)) + 2)) >= (( lower_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2))))) by A1, A237, A323, A329, Th19, XREAL_1: 64;

                          then ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) >= ((( lower_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 1))))) + (( lower_bound ( rng f)) * ( vol ( divset (D2,(( indx (D2,D1,n1)) + 2)))))) by A337, A335, XREAL_1: 7;

                          hence thesis by A279, A340, A326, A328;

                        end;

                      end;

                      hence thesis;

                    end;

                    

                     A341: (n1 + 1) <= ( len UVf(D1)) by A265, INTEGRA1:def 6;

                    ((j -' (n1 + 1)) + 1) = ((j - (n1 + 1)) + 1) by A279, XREAL_1: 233;

                    then

                     A342: ( len ( mid ( UVf(D1),(n1 + 1),j))) = 1 by A244, A279, A341, FINSEQ_6: 118;

                    reconsider uv = (( upper_bound ( rng (f | ( divset (D1,(n1 + 1)))))) * ( vol ( divset (D1,(n1 + 1))))) as Element of REAL by XREAL_0:def 1;

                    ((n1 + 1) + 1) <= (j + 1) by A258, XREAL_1: 6;

                    then 1 <= ((j + 1) - (n1 + 1)) by XREAL_1: 19;

                    

                    then (( mid ( UVf(D1),(n1 + 1),j)) . 1) = ( UVf(D1) . ((1 + (n1 + 1)) - 1)) by A244, A279, A341, FINSEQ_6: 122

                    .= (( upper_bound ( rng (f | ( divset (D1,(n1 + 1)))))) * ( vol ( divset (D1,(n1 + 1))))) by A266, INTEGRA1:def 6;

                    then ( mid ( UVf(D1),(n1 + 1),j)) = <*uv*> by A342, FINSEQ_1: 40;

                    then

                     A343: ( Sum ( mid ( UVf(D1),(n1 + 1),j))) = (( upper_bound ( rng (f | ( divset (D1,(n1 + 1)))))) * ( vol ( divset (D1,(n1 + 1))))) by FINSOP_1: 11;

                    ( divset (D1,(n1 + 1))) c= A by A266, INTEGRA1: 8;

                    then

                     A344: ( upper_bound ( rng (f | ( divset (D1,(n1 + 1)))))) <= ( upper_bound ( rng f)) by A1, Lm4;

                    (n1 + 1) in ( Seg ( len D1)) by A266, FINSEQ_1:def 3;

                    then (n1 + 1) in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by INTEGRA1:def 6;

                    then

                     A345: (n1 + 1) in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

                    ( vol ( divset (D1,(n1 + 1)))) = (( upper_volume (( chi (A,A)),D1)) . (n1 + 1)) by A266, INTEGRA1: 20;

                    then ( vol ( divset (D1,(n1 + 1)))) in ( rng ( upper_volume (( chi (A,A)),D1))) by A345, FUNCT_1:def 3;

                    then

                     A346: ( vol ( divset (D1,(n1 + 1)))) <= ( delta D1) by XXREAL_2:def 8;

                    (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                    then

                     A347: ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( vol ( divset (D1,(n1 + 1))))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1)) by A346, XREAL_1: 64;

                    ( vol ( divset (D1,(n1 + 1)))) >= 0 by INTEGRA1: 9;

                    then ( Sum ( mid ( UVf(D1),(n1 + 1),j))) <= (( upper_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1))))) by A343, A344, XREAL_1: 64;

                    then (( Sum ( mid ( UVf(D1),(n1 + 1),j))) - ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) <= ((( upper_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1))))) - (( lower_bound ( rng f)) * ( vol ( divset (D1,(n1 + 1)))))) by A315, XREAL_1: 13;

                    hence thesis by A347, XXREAL_0: 2;

                  end;

                    suppose

                     A348: (n1 + 1) < j;

                    

                     A349: (j -' (n1 + 1)) = (j - (n1 + 1)) by A258, XREAL_1: 233;

                    then

                     A350: ((j -' (n1 + 1)) + 1) = (j - n1);

                    

                     A351: n1 < (n1 + 1) by NAT_1: 13;

                    then

                     A352: (D1 . n1) < (D1 . (n1 + 1)) by A241, A266, SEQM_3:def 1;

                    then

                    consider B be non empty closed_interval Subset of REAL , MD1,MD2 be Division of B such that

                     A353: (D1 . n1) = ( lower_bound B) and ( upper_bound B) = (MD2 . ( len MD2)) and

                     A354: ( upper_bound B) = (MD1 . ( len MD1)) and

                     A355: MD1 <= MD2 and

                     A356: MD1 = ( mid (D1,(n1 + 1),j)) and

                     A357: MD2 = ( mid (D2,( indx (D2,D1,(n1 + 1))),( indx (D2,D1,j)))) by A10, A232, A258, A266, A276, Th15;

                    

                     A358: ( len MD1) = ((j -' (n1 + 1)) + 1) by A258, A261, A264, A244, A265, A356, FINSEQ_6: 118;

                    then

                     A359: ( len MD1) = ((j - (n1 + 1)) + 1) by A258, XREAL_1: 233;

                    then

                     A360: ((( len MD1) + (n1 + 1)) - 1) = j;

                    

                     A361: ( len MD1) in ( dom MD1) by FINSEQ_5: 6;

                    then

                     A362: 1 <= ( len MD1) by FINSEQ_3: 25;

                    

                     A363: ( lower_bound ( divset (MD1,( len MD1)))) = ( lower_bound ( divset (D1,j))) & ( upper_bound ( divset (MD1,( len MD1)))) = ( upper_bound ( divset (D1,j)))

                    proof

                      per cases ;

                        suppose

                         A364: ( len MD1) = 1;

                        then

                         A365: ( upper_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len MD1)) by A361, INTEGRA1:def 4;

                        

                         A366: ( upper_bound ( divset (D1,j))) = (D1 . j) by A232, A245, A277, INTEGRA1:def 4;

                        ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A232, A245, A277, INTEGRA1:def 4;

                        hence thesis by A261, A264, A353, A356, A359, A361, A364, A365, A366, FINSEQ_6: 118, INTEGRA1:def 4;

                      end;

                        suppose

                         A367: ( len MD1) <> 1;

                        then (( len MD1) - 1) in ( dom MD1) by A361, INTEGRA1: 7;

                        then

                         A368: (( len MD1) - 1) >= 1 by FINSEQ_3: 25;

                        ( len MD1) <= (( len MD1) + 1) by NAT_1: 11;

                        then

                         A369: (( len MD1) - 1) <= ( len MD1) by XREAL_1: 20;

                        ( upper_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len MD1)) by A361, A367, INTEGRA1:def 4;

                        then

                         A370: ( upper_bound ( divset (MD1,( len MD1)))) = (D1 . j) by A258, A264, A244, A356, A358, A360, A362, FINSEQ_6: 122;

                        

                         A371: (((( len MD1) - 1) + (n1 + 1)) - 1) = (j - 1) by A358, A349;

                        ( lower_bound ( divset (MD1,( len MD1)))) = (MD1 . (( len MD1) - 1)) by A361, A367, INTEGRA1:def 4;

                        then ( lower_bound ( divset (MD1,( len MD1)))) = (D1 . (j - 1)) by A258, A264, A244, A356, A358, A371, A368, A369, FINSEQ_6: 122;

                        hence thesis by A232, A245, A277, A370, INTEGRA1:def 4;

                      end;

                    end;

                    

                     A372: B c= A

                    proof

                      let x1 be object;

                      

                       A373: ( rng D1) c= A by INTEGRA1:def 2;

                      (D1 . n1) in ( rng D1) by A241, FUNCT_1:def 3;

                      then

                       A374: ( lower_bound A) <= (D1 . n1) by A373, INTEGRA2: 1;

                      assume

                       A375: x1 in B;

                      then

                      reconsider x1 as Real;

                      

                       A376: x1 <= (MD1 . ( len MD1)) by A354, A375, INTEGRA2: 1;

                      (D1 . j) in ( rng D1) by A232, FUNCT_1:def 3;

                      then

                       A377: (D1 . j) <= ( upper_bound A) by A373, INTEGRA2: 1;

                      (D1 . n1) <= x1 by A353, A375, INTEGRA2: 1;

                      then

                       A378: ( lower_bound A) <= x1 by A374, XXREAL_0: 2;

                      (MD1 . ( len MD1)) = (D1 . (((j - n1) - 1) + (n1 + 1))) by A258, A278, A264, A244, A356, A358, A349, FINSEQ_6: 122

                      .= (D1 . j);

                      then x1 <= ( upper_bound A) by A376, A377, XXREAL_0: 2;

                      hence thesis by A378, INTEGRA2: 1;

                    end;

                    then

                    reconsider g = (f | B) as Function of B, REAL by FUNCT_2: 32;

                    

                     A379: ( delta MD1) >= 0 by Th9;

                    

                     A380: (g | B) is bounded

                    proof

                      consider a be Real such that

                       A381: for x be object st x in (A /\ ( dom f)) holds a <= (f . x) by A1, RFUNCT_1: 71;

                      for x be object st x in (B /\ ( dom g)) holds a <= (g . x)

                      proof

                        let x be object;

                        

                         A382: (( dom f) /\ B) c= (( dom f) /\ A) by A372, XBOOLE_1: 26;

                        assume x in (B /\ ( dom g));

                        then

                         A383: x in ( dom g) by XBOOLE_0:def 4;

                        then x in (( dom f) /\ B) by RELAT_1: 61;

                        then a <= (f . x) by A381, A382;

                        hence thesis by A383, FUNCT_1: 47;

                      end;

                      then

                       A384: (g | B) is bounded_below by RFUNCT_1: 71;

                      consider a be Real such that

                       A385: for x be object st x in (A /\ ( dom f)) holds (f . x) <= a by A1, RFUNCT_1: 70;

                      for x be object st x in (B /\ ( dom g)) holds (g . x) <= a

                      proof

                        let x be object;

                        

                         A386: (( dom f) /\ B) c= (( dom f) /\ A) by A372, XBOOLE_1: 26;

                        assume x in (B /\ ( dom g));

                        then

                         A387: x in ( dom g) by XBOOLE_0:def 4;

                        then x in (( dom f) /\ B) by RELAT_1: 61;

                        then a >= (f . x) by A385, A386;

                        hence thesis by A387, FUNCT_1: 47;

                      end;

                      then (g | B) is bounded_above by RFUNCT_1: 70;

                      hence thesis by A384;

                    end;

                    ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A233, INTEGRA2: 1;

                    then

                     A388: (D1 . (j - 1)) <= (D . (i + 1)) by A232, A245, A277, INTEGRA1:def 4;

                    

                     A389: ((j -' (n1 + 1)) + 1) = ((j - (n1 + 1)) + 1) by A258, XREAL_1: 233;

                    

                     A390: ( len ( upper_volume (g,MD1))) = ( len MD1) by INTEGRA1:def 6

                    .= ((j - (n1 + 1)) + 1) by A258, A261, A264, A244, A265, A356, A389, FINSEQ_6: 118;

                    

                     A391: j <= ( len UVf(D1)) by A264, INTEGRA1:def 6;

                    

                     A392: for k be Nat st 1 <= k & k <= ( len ( upper_volume (g,MD1))) holds (( upper_volume (g,MD1)) . k) = (( mid ( UVf(D1),(n1 + 1),j)) . k)

                    proof

                      let k be Nat;

                      assume that

                       A393: 1 <= k and

                       A394: k <= ( len ( upper_volume (g,MD1)));

                      k in ( Seg ( len ( upper_volume (g,MD1)))) by A393, A394, FINSEQ_1: 1;

                      then

                       A395: k in ( Seg ( len MD1)) by INTEGRA1:def 6;

                      then

                       A396: k in ( dom MD1) by FINSEQ_1:def 3;

                      k in ( dom MD1) by A395, FINSEQ_1:def 3;

                      then

                       A397: (( upper_volume (g,MD1)) . k) = (( upper_bound ( rng (g | ( divset (MD1,k))))) * ( vol ( divset (MD1,k)))) by INTEGRA1:def 6;

                      consider k2 be Element of NAT such that

                       A398: (n1 + 1) = (1 + k2);

                      

                       A399: 1 <= (k + k2) by A393, NAT_1: 12;

                      k <= (j - ((n1 + 1) - 1)) by A390, A394;

                      then (k + ((n1 + 1) - 1)) <= j by XREAL_1: 19;

                      then (k + k2) <= ( len D1) by A264, A398, XXREAL_0: 2;

                      then

                       A400: (k + k2) in ( Seg ( len D1)) by A399, FINSEQ_1: 1;

                      then

                       A401: (k + k2) in ( dom D1) by FINSEQ_1:def 3;

                      (1 + 1) <= (k + k2) by A259, A393, A398, XREAL_1: 7;

                      then

                       A402: 1 < (k + k2) by NAT_1: 13;

                      

                       A403: k2 = ((n1 + 1) - 1) by A398;

                      

                       A404: ( lower_bound ( divset (D1,(k + k2)))) = ( lower_bound ( divset (MD1,k))) & ( upper_bound ( divset (D1,(k + k2)))) = ( upper_bound ( divset (MD1,k)))

                      proof

                        per cases ;

                          suppose

                           A405: k = 1;

                          then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A396, INTEGRA1:def 4;

                          then

                           A406: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A258, A264, A244, A356, A390, A393, A394, FINSEQ_6: 122;

                          ( lower_bound ( divset (MD1,k))) = (D1 . n1) by A353, A396, A405, INTEGRA1:def 4;

                          hence thesis by A403, A402, A401, A405, A406, INTEGRA1:def 4;

                        end;

                          suppose

                           A407: k <> 1;

                          then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A396, INTEGRA1:def 4;

                          then

                           A408: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A258, A264, A244, A356, A390, A393, A394, FINSEQ_6: 122;

                          

                           A409: (k - 1) <= ((j - (n1 + 1)) + 1) by A390, A394, XREAL_1: 146, XXREAL_0: 2;

                          

                           A410: ( lower_bound ( divset (MD1,k))) = (MD1 . (k - 1)) by A396, A407, INTEGRA1:def 4;

                          

                           A411: (k - 1) in ( dom MD1) by A396, A407, INTEGRA1: 7;

                          then 1 <= (k - 1) by FINSEQ_3: 25;

                          then ( lower_bound ( divset (MD1,k))) = (D1 . (((k - 1) + (n1 + 1)) - 1)) by A258, A264, A244, A356, A411, A409, A410, FINSEQ_6: 122;

                          hence thesis by A398, A402, A401, A408, INTEGRA1:def 4;

                        end;

                      end;

                      ( divset (MD1,k)) = [.( lower_bound ( divset (MD1,k))), ( upper_bound ( divset (MD1,k))).] by INTEGRA1: 4;

                      then

                       A412: ( divset (D1,(k + k2))) = ( divset (MD1,k)) by A404, INTEGRA1: 4;

                      

                       A413: (k + k2) in ( dom D1) by A400, FINSEQ_1:def 3;

                      

                       A414: (( mid ( UVf(D1),(n1 + 1),j)) . k) = ( UVf(D1) . ((k + (n1 + 1)) - 1)) by A258, A244, A390, A391, A393, A394, FINSEQ_6: 122

                      .= (( upper_bound ( rng (f | ( divset (D1,(k + k2)))))) * ( vol ( divset (D1,(k + k2))))) by A398, A413, INTEGRA1:def 6;

                      k in ( dom MD1) by A395, FINSEQ_1:def 3;

                      then ( divset (D1,(k + k2))) c= B by A412, INTEGRA1: 8;

                      hence thesis by A397, A414, A412, FUNCT_1: 51;

                    end;

                    (n1 + 1) <= ( len UVf(D1)) by A265, INTEGRA1:def 6;

                    then ( len ( upper_volume (g,MD1))) = ( len ( mid ( UVf(D1),(n1 + 1),j))) by A258, A261, A244, A389, A390, A391, FINSEQ_6: 118;

                    then

                     A415: ( Sum ( upper_volume (g,MD1))) = ( Sum ( mid ( UVf(D1),(n1 + 1),j))) by A392, FINSEQ_1: 14;

                    

                     A416: n1 < (j - 1) by A348, XREAL_1: 20;

                    

                     A417: 1 <= (( indx (D2,D1,n1)) + 1) by A263, NAT_1: 13;

                    

                     A418: ( len MD1) in ( dom MD1) by FINSEQ_5: 6;

                    

                     A419: ( upper_bound ( divset (MD1,( len MD1)))) = (MD1 . ( len MD1))

                    proof

                      per cases ;

                        suppose ( len MD1) = 1;

                        hence thesis by A418, INTEGRA1:def 4;

                      end;

                        suppose ( len MD1) <> 1;

                        hence thesis by A418, INTEGRA1:def 4;

                      end;

                    end;

                    ( vol B) = (( upper_bound B) - (D1 . n1)) by A353, INTEGRA1:def 5;

                    then ( vol B) = ((D1 . j) - (D1 . n1)) by A232, A245, A277, A354, A363, A419, INTEGRA1:def 4;

                    then

                     A420: ( vol B) <> 0 by A232, A241, A245, SEQM_3:def 1;

                    ( rng f) is bounded_below by A1, INTEGRA1: 11;

                    then

                     A421: ( lower_bound ( rng f)) <= ( lower_bound ( rng g)) by RELAT_1: 70, SEQ_4: 47;

                    ( rng f) is bounded_above by A1, INTEGRA1: 13;

                    then ( upper_bound ( rng f)) >= ( upper_bound ( rng g)) by RELAT_1: 70, SEQ_4: 48;

                    then (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= (( upper_bound ( rng g)) - ( lower_bound ( rng g))) by A421, XREAL_1: 13;

                    then

                     A422: ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) >= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta MD1)) by A379, XREAL_1: 64;

                    (D1 . n1) < (D1 . (n1 + 1)) by A241, A266, A351, SEQM_3:def 1;

                    then ( indx (D2,D1,n1)) < ( indx (D2,D1,(n1 + 1))) by A262, A260, A270, A267, SEQ_4: 137;

                    then

                     A423: (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,(n1 + 1))) by NAT_1: 13;

                    then

                     A424: (( indx (D2,D1,n1)) + 1) <= ( len D2) by A274, XXREAL_0: 2;

                    

                     A425: ( indx (D2,D1,n1)) < (( indx (D2,D1,n1)) + 1) by NAT_1: 13;

                    

                     A426: ( indx (D2,D1,(n1 + 1))) = (( indx (D2,D1,n1)) + 1)

                    proof

                      assume ( indx (D2,D1,(n1 + 1))) <> (( indx (D2,D1,n1)) + 1);

                      then

                       A427: ( indx (D2,D1,(n1 + 1))) > (( indx (D2,D1,n1)) + 1) by A423, XXREAL_0: 1;

                      

                       A428: (( indx (D2,D1,n1)) + 1) in ( dom D2) by A417, A424, FINSEQ_3: 25;

                      then

                       A429: (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D2) by FUNCT_1:def 3;

                      now

                        per cases by A11, A429, XBOOLE_0:def 3;

                          suppose (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D1);

                          then

                          consider n2 be Element of NAT such that

                           A430: n2 in ( dom D1) and

                           A431: (D2 . (( indx (D2,D1,n1)) + 1)) = (D1 . n2) by PARTFUN1: 3;

                          (D2 . ( indx (D2,D1,n1))) < (D2 . (( indx (D2,D1,n1)) + 1)) by A262, A425, A428, SEQM_3:def 1;

                          then n1 < n2 by A241, A260, A430, A431, SEQ_4: 137;

                          then

                           A432: (n1 + 1) <= n2 by NAT_1: 13;

                          (D1 . n2) < (D1 . (n1 + 1)) by A270, A267, A427, A428, A431, SEQM_3:def 1;

                          hence contradiction by A266, A430, A432, SEQ_4: 137;

                        end;

                          suppose

                           A433: (D2 . (( indx (D2,D1,n1)) + 1)) in ( rng D);

                          

                           A434: (D . i) <= ( upper_bound ( divset (D1,n1))) by A242, INTEGRA2: 1;

                          

                           A435: ( upper_bound ( divset (D1,n1))) = (D1 . n1)

                          proof

                            per cases ;

                              suppose n1 = 1;

                              hence thesis by A241, INTEGRA1:def 4;

                            end;

                              suppose n1 <> 1;

                              hence thesis by A241, INTEGRA1:def 4;

                            end;

                          end;

                          consider n2 be Element of NAT such that

                           A436: n2 in ( dom D) and

                           A437: (D2 . (( indx (D2,D1,n1)) + 1)) = (D . n2) by A433, PARTFUN1: 3;

                          (D1 . n1) < (D . n2) by A262, A260, A425, A428, A437, SEQM_3:def 1;

                          then (D . i) < (D . n2) by A434, A435, XXREAL_0: 2;

                          then i < n2 by A236, A436, SEQ_4: 137;

                          then

                           A438: (i + 1) <= n2 by NAT_1: 13;

                          ((n1 + 1) + 1) <= j by A348, NAT_1: 13;

                          then

                           A439: (n1 + 1) <= (j - 1) by XREAL_1: 19;

                          (j - 1) in ( dom D1) by A232, A245, A277, INTEGRA1: 7;

                          then

                           A440: (D1 . (n1 + 1)) <= (D1 . (j - 1)) by A266, A439, SEQ_4: 137;

                          

                           A441: ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A233, INTEGRA2: 1;

                          ( lower_bound ( divset (D1,j))) = (D1 . (j - 1)) by A232, A245, A277, INTEGRA1:def 4;

                          then

                           A442: (D1 . (n1 + 1)) <= (D . (i + 1)) by A440, A441, XXREAL_0: 2;

                          (D . n2) < (D1 . (n1 + 1)) by A270, A267, A427, A428, A437, SEQM_3:def 1;

                          then (D . n2) < (D . (i + 1)) by A442, XXREAL_0: 2;

                          hence contradiction by A231, A436, A438, SEQ_4: 137;

                        end;

                      end;

                      hence contradiction;

                    end;

                    

                     A443: ( len MD2) = ((( indx (D2,D1,j)) -' ( indx (D2,D1,(n1 + 1)))) + 1) by A275, A271, A274, A238, A239, A357, FINSEQ_6: 118;

                    then

                     A444: ( len MD2) = ((( indx (D2,D1,j)) - ( indx (D2,D1,(n1 + 1)))) + 1) by A275, XREAL_1: 233;

                    then

                     A445: ( len ( upper_volume (g,MD2))) = ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A426, INTEGRA1:def 6;

                    for x1 be object holds x1 in (( rng MD1) \/ {(D . (i + 1))}) implies x1 in ( rng MD2)

                    proof

                      let x1 be object;

                      assume

                       A446: x1 in (( rng MD1) \/ {(D . (i + 1))});

                      then

                      reconsider x1 as Real;

                      now

                        per cases by A446, XBOOLE_0:def 3;

                          suppose

                           A447: x1 in ( rng MD1);

                          ( rng MD1) <> {} ;

                          then 1 in ( dom MD1) by FINSEQ_3: 32;

                          then

                           A448: 1 <= ( len MD1) by FINSEQ_3: 25;

                          

                           A449: ( len MD1) = ((j -' (n1 + 1)) + 1) by A258, A261, A264, A244, A265, A356, FINSEQ_6: 118;

                          

                          then ((( len MD1) + (n1 + 1)) - 1) = ((((j - (n1 + 1)) + 1) + (n1 + 1)) - 1) by A258, XREAL_1: 233

                          .= j;

                          then

                           A450: (MD1 . ( len MD1)) = (D1 . j) by A258, A264, A244, A356, A448, A449, FINSEQ_6: 122;

                          ( rng MD1) c= ( rng D1) by A356, FINSEQ_6: 119;

                          then

                           A451: x1 in ( rng D1) by A447;

                          ( rng D1) c= ( rng D2) by A10, INTEGRA1:def 18;

                          then

                          consider k such that

                           A452: k in ( dom D2) and

                           A453: (D2 . k) = x1 by A451, PARTFUN1: 3;

                          x1 <= (MD1 . ( len MD1)) by A447, Th16;

                          then k <= ( indx (D2,D1,j)) by A237, A234, A450, A452, A453, SEQM_3:def 1;

                          then (k - ( indx (D2,D1,(n1 + 1)))) <= (( indx (D2,D1,j)) - ( indx (D2,D1,(n1 + 1)))) by XREAL_1: 9;

                          then

                           A454: ((k - ( indx (D2,D1,(n1 + 1)))) + 1) <= ((( indx (D2,D1,j)) - ( indx (D2,D1,(n1 + 1)))) + 1) by XREAL_1: 6;

                          

                           A455: (MD1 . 1) <= x1 by A447, Th16;

                          (MD1 . 1) = (D1 . (n1 + 1)) by A261, A264, A244, A265, A356, FINSEQ_6: 118;

                          then

                           A456: ( indx (D2,D1,(n1 + 1))) <= k by A270, A267, A455, A452, A453, SEQM_3:def 1;

                          then

                          consider n be Nat such that

                           A457: (k + 1) = (( indx (D2,D1,(n1 + 1))) + n) by NAT_1: 10, NAT_1: 12;

                          

                           A458: ((n + ( indx (D2,D1,(n1 + 1)))) - 1) = k by A457;

                          (( indx (D2,D1,(n1 + 1))) + 1) <= (k + 1) by A456, XREAL_1: 6;

                          then

                           A459: 1 <= ((k + 1) - ( indx (D2,D1,(n1 + 1)))) by XREAL_1: 19;

                          then n in ( dom MD2) by A444, A454, A457, FINSEQ_3: 25;

                          then (MD2 . n) in ( rng MD2) by FUNCT_1:def 3;

                          hence x1 in ( rng MD2) by A275, A271, A239, A357, A453, A459, A454, A458, FINSEQ_6: 122;

                        end;

                          suppose x1 in {(D . (i + 1))};

                          then

                           A460: x1 = (D . (i + 1)) by TARSKI:def 1;

                          reconsider j1 = (j - 1) as Element of NAT by A232, A245, A277, INTEGRA1: 7;

                          

                           A461: ( rng D) c= ( rng D2) by A9, INTEGRA1:def 18;

                          (D . (i + 1)) in ( rng D) by A231, FUNCT_1:def 3;

                          then

                          consider k such that

                           A462: k in ( dom D2) and

                           A463: x1 = (D2 . k) by A460, A461, PARTFUN1: 3;

                          (D . (i + 1)) <= ( upper_bound ( divset (D1,j))) by A233, INTEGRA2: 1;

                          then x1 <= (D1 . j) by A232, A245, A277, A460, INTEGRA1:def 4;

                          then

                           A464: (D2 . k) <= (D2 . ( indx (D2,D1,j))) by A10, A232, A463, INTEGRA1:def 19;

                          n1 < j1 by A348, XREAL_1: 20;

                          then

                           A465: (n1 + 1) <= j1 by NAT_1: 13;

                          (j - 1) in ( dom D1) by A232, A245, A277, INTEGRA1: 7;

                          then

                           A466: (D1 . (n1 + 1)) <= (D1 . (j - 1)) by A266, A465, SEQ_4: 137;

                          ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A233, INTEGRA2: 1;

                          then (D1 . (j - 1)) <= x1 by A232, A245, A277, A460, INTEGRA1:def 4;

                          then (D2 . ( indx (D2,D1,(n1 + 1)))) <= (D2 . k) by A267, A463, A466, XXREAL_0: 2;

                          hence x1 in ( rng MD2) by A270, A237, A357, A462, A463, A464, Th17;

                        end;

                      end;

                      hence thesis;

                    end;

                    then

                     A467: (( rng MD1) \/ {(D . (i + 1))}) c= ( rng MD2);

                    ( rng MD2) <> {} ;

                    then 1 in ( dom MD2) by FINSEQ_3: 32;

                    then

                     A468: 1 <= ( len MD2) by FINSEQ_3: 25;

                    

                     A469: ((( len MD2) - 1) + ( indx (D2,D1,(n1 + 1)))) = ( indx (D2,D1,j)) by A444;

                    for x1 be object holds x1 in ( rng MD2) implies x1 in (( rng MD1) \/ {(D . (i + 1))})

                    proof

                      let x1 be object;

                      assume

                       A470: x1 in ( rng MD2);

                      then

                      reconsider x1 as Real;

                      

                       A471: (MD2 . 1) <= x1 by A470, Th16;

                      

                       A472: (MD2 . ( len MD2)) = (D2 . ( indx (D2,D1,j))) by A275, A271, A239, A357, A468, A443, A469, FINSEQ_6: 122;

                      

                       A473: ( rng MD2) c= ( rng D2) by A357, FINSEQ_6: 119;

                      

                       A474: (MD2 . 1) = (D2 . ( indx (D2,D1,(n1 + 1)))) by A271, A274, A238, A239, A357, FINSEQ_6: 118;

                      

                       A475: x1 <= (MD2 . ( len MD2)) by A470, Th16;

                      then

                       A476: x1 <= (D1 . j) by A234, A275, A271, A239, A357, A468, A443, A469, FINSEQ_6: 122;

                      now

                        per cases by A11, A470, A473, XBOOLE_0:def 3;

                          suppose x1 in ( rng D1);

                          then

                          consider k such that

                           A477: k in ( dom D1) and

                           A478: (D1 . k) = x1 by PARTFUN1: 3;

                          

                           A479: (n1 + 1) <= k by A266, A267, A471, A474, A477, A478, SEQM_3:def 1;

                          then

                           A480: 1 <= (k - n1) by XREAL_1: 19;

                          n1 <= (n1 + 1) by NAT_1: 11;

                          then

                          consider n be Nat such that

                           A481: k = (n1 + n) by A479, NAT_1: 10, XXREAL_0: 2;

                          

                           A482: k <= j by A232, A234, A475, A472, A477, A478, SEQM_3:def 1;

                          then

                           A483: (k - n1) <= (j - n1) by XREAL_1: 9;

                          

                           A484: 1 <= (k - n1) by A479, XREAL_1: 19;

                          

                           A485: ((j - (n1 + 1)) + 1) = (j - n1);

                          (k - n1) <= ( len MD1) by A358, A350, A482, XREAL_1: 9;

                          then n in ( dom MD1) by A484, A481, FINSEQ_3: 25;

                          then

                           A486: (MD1 . n) in ( rng MD1) by FUNCT_1:def 3;

                          (MD1 . n) = (D1 . (((k - n1) - 1) + (n1 + 1))) by A258, A264, A244, A356, A480, A483, A485, A481, FINSEQ_6: 122

                          .= (D1 . k);

                          hence x1 in (( rng MD1) \/ {(D . (i + 1))}) by A478, A486, XBOOLE_0:def 3;

                        end;

                          suppose x1 in ( rng D);

                          then

                          consider n such that

                           A487: n in ( dom D) and

                           A488: (D . n) = x1 by PARTFUN1: 3;

                          

                           A489: not (i + 1) < n

                          proof

                            j = 1 or j <> 1;

                            then

                             A490: ( upper_bound ( divset (D1,j))) = (D1 . j) by A232, INTEGRA1:def 4;

                            reconsider y1 = (D . (i + 1)) as Real;

                            

                             A491: (D . n) in ( rng D) by A487, FUNCT_1:def 3;

                            assume (i + 1) < n;

                            then

                             A492: (D . (i + 1)) < (D . n) by A231, A487, SEQM_3:def 1;

                            ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A233, INTEGRA2: 1;

                            then ( lower_bound ( divset (D1,j))) <= (D . n) by A492, XXREAL_0: 2;

                            then (D . n) in ( divset (D1,j)) by A476, A488, A490, INTEGRA2: 1;

                            then

                             A493: x1 in (( rng D) /\ ( divset (D1,j))) by A488, A491, XBOOLE_0:def 4;

                            (D . (i + 1)) in ( rng D) by A231, FUNCT_1:def 3;

                            then y1 in (( rng D) /\ ( divset (D1,j))) by A233, XBOOLE_0:def 4;

                            hence contradiction by A8, A232, A488, A492, A493, Th5;

                          end;

                          

                           A494: ( upper_bound ( divset (D1,n1))) = (D1 . n1)

                          proof

                            per cases ;

                              suppose n1 = 1;

                              hence thesis by A241, INTEGRA1:def 4;

                            end;

                              suppose n1 <> 1;

                              hence thesis by A241, INTEGRA1:def 4;

                            end;

                          end;

                          (D . i) <= ( upper_bound ( divset (D1,n1))) by A242, INTEGRA2: 1;

                          then (D . i) < (D1 . (n1 + 1)) by A352, A494, XXREAL_0: 2;

                          then (D . i) < (D . n) by A267, A471, A474, A488, XXREAL_0: 2;

                          then i < n by A236, A487, SEQ_4: 137;

                          then (i + 1) <= n by NAT_1: 13;

                          then (i + 1) = n or (i + 1) < n by XXREAL_0: 1;

                          then x1 in {(D . (i + 1))} by A488, A489, TARSKI:def 1;

                          hence x1 in (( rng MD1) \/ {(D . (i + 1))}) by XBOOLE_0:def 3;

                        end;

                      end;

                      hence thesis;

                    end;

                    then ( rng MD2) c= (( rng MD1) \/ {(D . (i + 1))});

                    then

                     A495: ( rng MD2) = (( rng MD1) \/ {(D . (i + 1))}) by A467, XBOOLE_0:def 10;

                    ( delta MD1) in ( rng ( upper_volume (( chi (B,B)),MD1))) by XXREAL_2:def 8;

                    then

                    consider k such that

                     A496: k in ( dom ( upper_volume (( chi (B,B)),MD1))) and

                     A497: (( upper_volume (( chi (B,B)),MD1)) . k) = ( delta MD1) by PARTFUN1: 3;

                    

                     A498: k in ( Seg ( len ( upper_volume (( chi (B,B)),MD1)))) by A496, FINSEQ_1:def 3;

                    then

                     A499: k in ( Seg ( len MD1)) by INTEGRA1:def 6;

                    then

                     A500: k in ( dom MD1) by FINSEQ_1:def 3;

                    

                     A501: k <= ( len MD1) by A499, FINSEQ_1: 1;

                    then (k + n1) <= j by A358, A350, XREAL_1: 19;

                    then

                     A502: (k + n1) <= ( len D1) by A264, XXREAL_0: 2;

                    

                     A503: 1 <= k by A498, FINSEQ_1: 1;

                    

                     A504: (n1 + 1) > 1 by A277, NAT_1: 13;

                    then n1 > (1 - 1) by XREAL_1: 19;

                    then

                     A505: k < (k + n1) by XREAL_1: 29;

                    then 1 < (k + n1) by A503, XXREAL_0: 2;

                    then

                     A506: (k + n1) in ( dom D1) by A502, FINSEQ_3: 25;

                    ( lower_bound ( divset (MD1,k))) = ( lower_bound ( divset (D1,(k + n1)))) & ( upper_bound ( divset (MD1,k))) = ( upper_bound ( divset (D1,(k + n1))))

                    proof

                      per cases ;

                        suppose

                         A507: k = 1;

                        then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A500, INTEGRA1:def 4;

                        then

                         A508: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A258, A264, A244, A356, A358, A503, A501, FINSEQ_6: 122;

                        ( lower_bound ( divset (D1,(k + n1)))) = (D1 . ((k + n1) - 1)) by A503, A505, A506, INTEGRA1:def 4;

                        hence thesis by A353, A504, A500, A506, A507, A508, INTEGRA1:def 4;

                      end;

                        suppose

                         A509: k <> 1;

                        then ( upper_bound ( divset (MD1,k))) = (MD1 . k) by A500, INTEGRA1:def 4;

                        then

                         A510: ( upper_bound ( divset (MD1,k))) = (D1 . ((k + (n1 + 1)) - 1)) by A258, A264, A244, A356, A358, A503, A501, FINSEQ_6: 122;

                        

                         A511: ( lower_bound ( divset (MD1,k))) = (MD1 . (k - 1)) by A500, A509, INTEGRA1:def 4;

                        

                         A512: (k - 1) in ( dom MD1) by A500, A509, INTEGRA1: 7;

                        then

                         A513: (k - 1) <= ( len MD1) by FINSEQ_3: 25;

                        1 <= (k - 1) by A512, FINSEQ_3: 25;

                        then ( lower_bound ( divset (MD1,k))) = (D1 . (((k - 1) + (n1 + 1)) - 1)) by A258, A264, A244, A356, A358, A512, A513, A511, FINSEQ_6: 122;

                        hence thesis by A503, A505, A506, A510, INTEGRA1:def 4;

                      end;

                    end;

                    then ( divset (MD1,k)) = [.( lower_bound ( divset (D1,(k + n1)))), ( upper_bound ( divset (D1,(k + n1)))).] by INTEGRA1: 4;

                    then

                     A514: ( divset (MD1,k)) = ( divset (D1,(k + n1))) by INTEGRA1: 4;

                    (k + n1) in ( Seg ( len D1)) by A506, FINSEQ_1:def 3;

                    then (k + n1) in ( Seg ( len ( upper_volume (( chi (A,A)),D1)))) by INTEGRA1:def 6;

                    then

                     A515: (k + n1) in ( dom ( upper_volume (( chi (A,A)),D1))) by FINSEQ_1:def 3;

                    k in ( dom MD1) by A499, FINSEQ_1:def 3;

                    then ( delta MD1) = ( vol ( divset (MD1,k))) by A497, INTEGRA1: 20;

                    then ( delta MD1) = (( upper_volume (( chi (A,A)),D1)) . (k + n1)) by A506, A514, INTEGRA1: 20;

                    then ( delta MD1) in ( rng ( upper_volume (( chi (A,A)),D1))) by A515, FUNCT_1:def 3;

                    then ( delta MD1) <= ( max ( rng ( upper_volume (( chi (A,A)),D1)))) by XXREAL_2:def 8;

                    then

                     A516: ( delta MD1) <= ( delta D1);

                    

                     A517: (D . (i + 1)) <= ( upper_bound ( divset (D1,j))) by A233, INTEGRA2: 1;

                    ( lower_bound ( divset (D1,j))) <= (D . (i + 1)) by A233, INTEGRA2: 1;

                    then

                     A518: (D . (i + 1)) in ( divset (MD1,( len MD1))) by A363, A517, INTEGRA2: 1;

                    (j - 1) in ( dom D1) by A232, A245, A277, INTEGRA1: 7;

                    then (D1 . n1) < (D1 . (j - 1)) by A241, A416, SEQM_3:def 1;

                    then (D . (i + 1)) > ( lower_bound B) by A353, A388, XXREAL_0: 2;

                    then (( Sum ( upper_volume (g,MD1))) - ( Sum ( upper_volume (g,MD2)))) <= ((( upper_bound ( rng g)) - ( lower_bound ( rng g))) * ( delta MD1)) by A355, A380, A495, A518, A420, Th14;

                    then

                     A519: (( Sum ( upper_volume (g,MD1))) - ( Sum ( upper_volume (g,MD2)))) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) by A422, XXREAL_0: 2;

                    

                     A520: ( indx (D2,D1,j)) <= ( len UVf(D2)) by A239, INTEGRA1:def 6;

                    

                     A521: (( indx (D2,D1,n1)) + 1) <= ( indx (D2,D1,j)) by A275, A423, XXREAL_0: 2;

                    

                     A522: for k be Nat st 1 <= k & k <= ( len ( upper_volume (g,MD2))) holds (( upper_volume (g,MD2)) . k) = (( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . k)

                    proof

                      let k be Nat;

                      assume that

                       A523: 1 <= k and

                       A524: k <= ( len ( upper_volume (g,MD2)));

                      

                       A525: k in ( Seg ( len ( upper_volume (g,MD2)))) by A523, A524, FINSEQ_1: 1;

                      

                       A526: (( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . k) = ( UVf(D2) . ((k + (( indx (D2,D1,n1)) + 1)) - 1)) by A417, A445, A520, A521, A523, A524, FINSEQ_6: 122;

                      

                       A527: k in ( Seg ( len MD2)) by A525, INTEGRA1:def 6;

                      then k in ( dom MD2) by FINSEQ_1:def 3;

                      then

                       A528: (( upper_volume (g,MD2)) . k) = (( upper_bound ( rng (g | ( divset (MD2,k))))) * ( vol ( divset (MD2,k)))) by INTEGRA1:def 6;

                      1 <= (( indx (D2,D1,n1)) + 1) by NAT_1: 12;

                      then (1 + 1) <= (k + (( indx (D2,D1,n1)) + 1)) by A523, XREAL_1: 7;

                      then

                       A529: 1 <= ((k + (( indx (D2,D1,n1)) + 1)) - 1) by XREAL_1: 19;

                      consider k2 be Element of NAT such that

                       A530: (( indx (D2,D1,n1)) + 1) = (1 + k2);

                      k <= (( indx (D2,D1,j)) - ((( indx (D2,D1,n1)) + 1) - 1)) by A444, A426, A524, INTEGRA1:def 6;

                      then (k + ((( indx (D2,D1,n1)) + 1) - 1)) <= ( indx (D2,D1,j)) by XREAL_1: 19;

                      then ((k + (( indx (D2,D1,n1)) + 1)) - 1) <= ( len UVf(D2)) by A520, XXREAL_0: 2;

                      then (k + k2) in ( Seg ( len UVf(D2))) by A529, A530, FINSEQ_1: 1;

                      then

                       A531: (k + k2) in ( Seg ( len D2)) by INTEGRA1:def 6;

                      then (k + k2) in ( dom D2) by FINSEQ_1:def 3;

                      then

                       A532: (( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))) . k) = (( upper_bound ( rng (f | ( divset (D2,(k + k2)))))) * ( vol ( divset (D2,(k + k2))))) by A526, A530, INTEGRA1:def 6;

                      

                       A533: ( lower_bound ( divset (MD2,k))) = ( lower_bound ( divset (D2,(k + k2)))) & ( upper_bound ( divset (MD2,k))) = ( upper_bound ( divset (D2,(k + k2))))

                      proof

                        (k + k2) >= (1 + 1) by A263, A523, A530, XREAL_1: 7;

                        then

                         A534: (k + k2) > 1 by NAT_1: 13;

                        

                         A535: k in ( dom MD2) by A527, FINSEQ_1:def 3;

                        

                         A536: (k + k2) in ( dom D2) by A531, FINSEQ_1:def 3;

                        per cases ;

                          suppose

                           A537: k = 1;

                          then

                           A538: ( upper_bound ( divset (D2,(k + k2)))) = (D2 . (1 + k2)) by A534, A536, INTEGRA1:def 4;

                          

                           A539: ( lower_bound ( divset (MD2,k))) = ( lower_bound B) by A535, A537, INTEGRA1:def 4;

                          ( upper_bound ( divset (MD2,k))) = (MD2 . k) by A535, A537, INTEGRA1:def 4;

                          

                          then

                           A540: ( upper_bound ( divset (MD2,k))) = (D2 . ((1 + ( indx (D2,D1,(n1 + 1)))) - 1)) by A275, A239, A357, A417, A426, A445, A524, A537, FINSEQ_6: 122

                          .= (D1 . (n1 + 1)) by A10, A266, INTEGRA1:def 19;

                          ( lower_bound ( divset (D2,(k + k2)))) = (D2 . ((1 + k2) - 1)) by A534, A536, A537, INTEGRA1:def 4;

                          hence thesis by A10, A241, A266, A353, A426, A530, A539, A540, A538, INTEGRA1:def 19;

                        end;

                          suppose

                           A541: k <> 1;

                          then ( upper_bound ( divset (MD2,k))) = (MD2 . k) by A535, INTEGRA1:def 4;

                          then

                           A542: ( upper_bound ( divset (MD2,k))) = (D2 . ((k + (( indx (D2,D1,n1)) + 1)) - 1)) by A275, A239, A357, A417, A426, A445, A523, A524, FINSEQ_6: 122;

                          

                           A543: (k - 1) <= ((( indx (D2,D1,j)) - (( indx (D2,D1,n1)) + 1)) + 1) by A445, A524, XREAL_1: 146, XXREAL_0: 2;

                          

                           A544: ( lower_bound ( divset (MD2,k))) = (MD2 . (k - 1)) by A535, A541, INTEGRA1:def 4;

                          

                           A545: (k - 1) in ( dom MD2) by A535, A541, INTEGRA1: 7;

                          then 1 <= (k - 1) by FINSEQ_3: 25;

                          then ( lower_bound ( divset (MD2,k))) = (D2 . (((k - 1) + (( indx (D2,D1,n1)) + 1)) - 1)) by A275, A239, A357, A417, A426, A545, A543, A544, FINSEQ_6: 122;

                          hence thesis by A530, A534, A536, A542, INTEGRA1:def 4;

                        end;

                      end;

                      ( divset (MD2,k)) = [.( lower_bound ( divset (MD2,k))), ( upper_bound ( divset (MD2,k))).] by INTEGRA1: 4;

                      then

                       A546: ( divset (MD2,k)) = ( divset (D2,(k + k2))) by A533, INTEGRA1: 4;

                      k in ( dom MD2) by A527, FINSEQ_1:def 3;

                      then ( divset (D2,(k + k2))) c= B by A546, INTEGRA1: 8;

                      hence thesis by A528, A532, A546, FUNCT_1: 51;

                    end;

                    (( indx (D2,D1,n1)) + 1) <= ( len UVf(D2)) by A424, INTEGRA1:def 6;

                    then ( len ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) = ((( indx (D2,D1,j)) -' (( indx (D2,D1,n1)) + 1)) + 1) by A238, A417, A520, A521, FINSEQ_6: 118;

                    then ( len ( upper_volume (g,MD2))) = ( len ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) by A275, A423, A445, XREAL_1: 233, XXREAL_0: 2;

                    then

                     A547: ( Sum ( upper_volume (g,MD2))) = ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))) by A522, FINSEQ_1: 14;

                    (( upper_bound ( rng f)) - ( lower_bound ( rng f))) >= 0 by A1, Lm3, XREAL_1: 48;

                    then ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta MD1)) <= ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1)) by A516, XREAL_1: 64;

                    hence thesis by A519, A547, A415, XXREAL_0: 2;

                  end;

                end;

                then

                 A548: (( PUf(D1,n1) - PUf(D2,indx)) + (( Sum ( mid ( UVf(D1),(n1 + 1),j))) - ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))))) <= (((i * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) + ((( upper_bound ( rng f)) - ( lower_bound ( rng f))) * ( delta D1))) by A243, XREAL_1: 7;

                n1 < (n1 + 1) by NAT_1: 13;

                then (D1 . n1) < (D1 . (n1 + 1)) by A241, A266, SEQM_3:def 1;

                then ( indx (D2,D1,n1)) < ( indx (D2,D1,(n1 + 1))) by A262, A260, A270, A267, SEQ_4: 137;

                then

                 A549: ( indx (D2,D1,n1)) < ( indx (D2,D1,j)) by A275, XXREAL_0: 2;

                ( indx (D2,D1,n1)) in ( Seg ( len D2)) by A262, FINSEQ_1:def 3;

                then ( indx (D2,D1,n1)) in ( Seg ( len UVf(D2))) by INTEGRA1:def 6;

                then ( indx (D2,D1,n1)) in ( dom UVf(D2)) by FINSEQ_1:def 3;

                

                then PUf(D2,indx) = ( Sum ( UVf(D2) | ( indx (D2,D1,n1)))) by INTEGRA1:def 20

                .= ( Sum ( mid ( UVf(D2),1,( indx (D2,D1,n1))))) by A263, FINSEQ_6: 116;

                

                then ( PUf(D2,indx) + ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) = ( Sum (( mid ( UVf(D2),1,( indx (D2,D1,n1)))) ^ ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) by RVSUM_1: 75

                .= ( Sum ( mid ( UVf(D2),1,( indx (D2,D1,j))))) by A263, A549, A240, INTEGRA2: 4

                .= ( Sum ( UVf(D2) | ( indx (D2,D1,j)))) by A238, FINSEQ_6: 116;

                then PUf(D2,indx) = ( PUf(D2,indx) + ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j)))))) by A273, INTEGRA1:def 20;

                then (( PUf(D1,n1) - PUf(D2,indx)) + (( Sum ( mid ( UVf(D1),(n1 + 1),j))) - ( Sum ( mid ( UVf(D2),(( indx (D2,D1,n1)) + 1),( indx (D2,D1,j))))))) = ( PUf(D1,j) - PUf(D2,indx)) by A272;

                hence thesis by A232, A233, A548;

              end;

              hence thesis;

            end;

            for k be non zero Nat holds P[k] from NAT_1:sch 10( A36, A227);

            then P[i];

            hence thesis by A14;

          end;

          

           A550: ( len D1) in ( dom D1) by FINSEQ_5: 6;

          then (D1 . ( len D1)) = (D2 . ( indx (D2,D1,( len D1)))) by A10, INTEGRA1:def 19;

          then ( upper_bound A) = (D2 . ( indx (D2,D1,( len D1)))) by INTEGRA1:def 2;

          then

           A551: (D2 . ( len D2)) = (D2 . ( indx (D2,D1,( len D1)))) by INTEGRA1:def 2;

          ( len D) in ( dom D) by FINSEQ_5: 6;

          then

          consider j such that

           A552: j in ( dom D1) and

           A553: (D . ( len D)) in ( divset (D1,j)) and

           A554: ( PUf(D1,j) - PUf(D2,indx)) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A13;

          

           A555: j = ( len D1)

          proof

            j in ( Seg ( len D1)) by A552, FINSEQ_1:def 3;

            then

             A556: j <= ( len D1) by FINSEQ_1: 1;

            assume j <> ( len D1);

            then j < ( len D1) by A556, XXREAL_0: 1;

            then (D1 . j) < (D1 . ( len D1)) by A552, A550, SEQM_3:def 1;

            then

             A557: (D1 . j) < ( upper_bound A) by INTEGRA1:def 2;

            

             A558: ( upper_bound ( divset (D1,j))) < ( upper_bound A)

            proof

              per cases ;

                suppose j = 1;

                hence thesis by A552, A557, INTEGRA1:def 4;

              end;

                suppose j <> 1;

                hence thesis by A552, A557, INTEGRA1:def 4;

              end;

            end;

            (D . ( len D)) <= ( upper_bound ( divset (D1,j))) by A553, INTEGRA2: 1;

            hence contradiction by A558, INTEGRA1:def 2;

          end;

          ( indx (D2,D1,( len D1))) in ( dom D2) by A10, A550, INTEGRA1:def 19;

          then ( indx (D2,D1,( len D1))) = ( len D2) by A12, A551, SEQ_4: 138;

          then (( upper_sum (f,D1)) - PUf(D2,len)) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A554, A555, INTEGRA1: 42;

          hence thesis by INTEGRA1: 42;

        end;

        hence thesis by A9, A10, A11;

      end;

      hence thesis;

    end;

    ::$Notion-Name

    theorem :: INTEGRA3:25

    (f | A) is bounded & ( delta T) is 0 -convergent non-zero & ( vol A) <> 0 implies ( lower_sum (f,T)) is convergent & ( lim ( lower_sum (f,T))) = ( lower_integral f)

    proof

      assume that

       A1: (f | A) is bounded and

       A2: ( delta T) is 0 -convergent non-zero and

       A3: ( vol A) <> 0 ;

      

       A4: ( delta T) is convergent by A2, FDIFF_1:def 1;

      

       A5: for D, D1 holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D))) & 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D1))) by A1, Th20;

      

       A10: for D, D1 st ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D)))) holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & (( lower_sum (f,D2)) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A1, Th21;

      

       A552: ( lim ( delta T)) = 0 by A2, FDIFF_1:def 1;

      

       A553: ( delta T) is non-zero by A2;

      

       A554: for e st e > 0 holds ex n be Nat st for m be Nat st n <= m holds 0 < (( delta T) . m) & (( delta T) . m) < e

      proof

        let e;

        assume e > 0 ;

        then

        consider n be Nat such that

         A555: for m be Nat st n <= m holds |.((( delta T) . m) - 0 ).| < e by A4, A552, SEQ_2:def 7;

        take n;

        let m be Nat;

        reconsider mm = m as Element of NAT by ORDINAL1:def 12;

        

         A556: (( delta T) . m) = ( delta (T . mm)) by Def2;

        ( delta (T . mm)) in ( rng ( upper_volume (( chi (A,A)),(T . mm)))) by XXREAL_2:def 8;

        then

        consider i such that

         A557: i in ( dom ( upper_volume (( chi (A,A)),(T . mm)))) and

         A558: ( delta (T . mm)) = (( upper_volume (( chi (A,A)),(T . mm))) . i) by PARTFUN1: 3;

        consider D be Division of A such that

         A559: D = (T . mm);

        i in ( Seg ( len ( upper_volume (( chi (A,A)),(T . mm))))) by A557, FINSEQ_1:def 3;

        then i in ( Seg ( len D)) by A559, INTEGRA1:def 6;

        then i in ( dom D) by FINSEQ_1:def 3;

        then

         A560: ( delta (T . mm)) = ( vol ( divset ((T . mm),i))) by A558, A559, INTEGRA1: 20;

        assume n <= m;

        then |.((( delta T) . m) - 0 ).| < e by A555;

        then

         A561: ((( delta T) . m) + |.((( delta T) . m) - 0 ).|) < (e + |.((( delta T) . m) - 0 ).|) by ABSVALUE: 4, XREAL_1: 8;

        (( delta T) . m) <> 0 by A553, SEQ_1: 5;

        hence thesis by A561, A556, A560, INTEGRA1: 9, XREAL_1: 6;

      end;

      

       A562: for e be Real st e > 0 holds ex n be Nat st for m be Nat st n <= m holds |.((( lower_sum (f,T)) . m) - ( lower_integral f)).| < e

      proof

        set h = ( lower_bound ( rng f));

        set H = ( upper_bound ( rng f));

        let e be Real;

        assume

         A563: e > 0 ;

        then

         A564: (e / 2) > 0 by XREAL_1: 139;

        reconsider e as Real;

        

         A565: (H - h) >= 0 by A1, Lm3, XREAL_1: 48;

        

         A566: ( rng ( lower_sum_set f)) is bounded_above by A1, INTEGRA2: 36;

        ( lower_integral f) = ( upper_bound ( rng ( lower_sum_set f))) by INTEGRA1:def 15;

        then

        consider y be Real such that

         A567: y in ( rng ( lower_sum_set f)) and

         A568: (( lower_integral f) - (e / 2)) < y by A564, A566, SEQ_4:def 1;

        consider D be Division of A such that D in ( dom ( lower_sum_set f)) and

         A569: y = (( lower_sum_set f) . D) and

         A570: (D . 1) > ( lower_bound A) by A3, A567, Lm7;

        deffunc F( Nat) = ( In (( vol ( divset (D,$1))), REAL ));

        set p = ( len D);

        consider v be FinSequence of REAL such that

         A571: ( len v) = ( len D) & for j be Nat st j in ( dom v) holds (v . j) = F(j) from FINSEQ_2:sch 1;

        consider v1 be non-decreasing FinSequence of REAL such that

         A572: (v,v1) are_fiberwise_equipotent by INTEGRA2: 3;

        defpred P[ Nat] means $1 in ( dom v1) & (v1 . $1) > 0 ;

        

         A573: ( dom v) = ( Seg ( len D)) by A571, FINSEQ_1:def 3;

        

         A574: ex k be Nat st P[k]

        proof

          consider H be Function such that ( dom H) = ( dom v) and ( rng H) = ( dom v1) and H is one-to-one and

           A575: v = (v1 * H) by A572, CLASSES1: 77;

          consider k such that

           A576: k in ( dom D) and

           A577: ( vol ( divset (D,k))) > 0 by A3, Th2;

          

           A578: ( dom D) = ( Seg ( len v)) by A571, FINSEQ_1:def 3;

          then (H . k) in ( dom v1) by A571, A573, A575, A576, FUNCT_1: 11;

          then

          reconsider Hk = (H . k) as Nat;

          (v . k) = F(k) by A571, A578, A573, A576;

          then (v . k) > 0 by A577;

          then P[Hk] by A571, A573, A575, A576, A578, FUNCT_1: 11, FUNCT_1: 12;

          hence thesis;

        end;

        consider k be Nat such that

         A579: P[k] & for n be Nat st P[n] holds k <= n from NAT_1:sch 5( A574);

        

         A580: (2 * p) > 0 by XREAL_1: 129;

        then

         A581: ((2 * p) * ((H - h) + 1)) > 0 by A565, XREAL_1: 129;

        ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) > 0

        proof

          per cases by XXREAL_0: 15;

            suppose ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) = (v1 . k);

            hence thesis by A579;

          end;

            suppose ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) = (e / ((2 * p) * ((H - h) + 1)));

            hence thesis by A563, A581, XREAL_1: 139;

          end;

        end;

        then

        consider n be Nat such that

         A582: for m be Nat st n <= m holds 0 < (( delta T) . m) & (( delta T) . m) < ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) by A554;

        take n;

        

         A583: y = ( lower_sum (f,D)) by A569, INTEGRA1:def 11;

        

         A584: (v1 . 1) > 0

        proof

          

           A585: for n1 be Element of NAT st n1 in ( dom D) holds ( vol ( divset (D,n1))) > 0

          proof

            let n1 be Element of NAT ;

            assume

             A586: n1 in ( dom D);

            then

             A587: 1 <= n1 by FINSEQ_3: 25;

            per cases by A587, XXREAL_0: 1;

              suppose

               A588: n1 = 1;

              then

               A589: ( upper_bound ( divset (D,n1))) = (D . n1) by A586, INTEGRA1:def 4;

              ( lower_bound ( divset (D,n1))) = ( lower_bound A) by A586, A588, INTEGRA1:def 4;

              then ( vol ( divset (D,n1))) = ((D . n1) - ( lower_bound A)) by A589, INTEGRA1:def 5;

              hence thesis by A570, A588, XREAL_1: 50;

            end;

              suppose

               A590: n1 > 1;

              then

               A591: ( upper_bound ( divset (D,n1))) = (D . n1) by A586, INTEGRA1:def 4;

              ( lower_bound ( divset (D,n1))) = (D . (n1 - 1)) by A586, A590, INTEGRA1:def 4;

              then

               A592: ( vol ( divset (D,n1))) = ((D . n1) - (D . (n1 - 1))) by A591, INTEGRA1:def 5;

              n1 < (n1 + 1) by XREAL_1: 29;

              then

               A593: (n1 - 1) < n1 by XREAL_1: 19;

              (n1 - 1) in ( dom D) by A586, A590, INTEGRA1: 7;

              then (D . (n1 - 1)) < (D . n1) by A586, A593, SEQM_3:def 1;

              hence thesis by A592, XREAL_1: 50;

            end;

          end;

          

           A594: k <= ( len v1) by A579, FINSEQ_3: 25;

          1 <= k by A579, FINSEQ_3: 25;

          then 1 <= ( len v1) by A594, XXREAL_0: 2;

          then 1 in ( dom v1) by FINSEQ_3: 25;

          then

           A595: (v1 . 1) in ( rng v1) by FUNCT_1:def 3;

          ( rng v) = ( rng v1) by A572, CLASSES1: 75;

          then

          consider n1 be Element of NAT such that

           A596: n1 in ( dom v) and

           A597: (v1 . 1) = (v . n1) by A595, PARTFUN1: 3;

          n1 in ( Seg ( len D)) by A571, A596, FINSEQ_1:def 3;

          then

           A598: n1 in ( dom D) by FINSEQ_1:def 3;

          (v1 . 1) = F(n1) by A571, A596, A597

          .= ( vol ( divset (D,n1)));

          hence thesis by A585, A598;

        end;

        

         A599: (v1 . k) = ( min ( rng ( upper_volume (( chi (A,A)),D))))

        proof

          

           A600: k = 1

          proof

            ( len v1) = ( len v) by A572, RFINSEQ: 3;

            then k in ( Seg ( len v)) by A579, FINSEQ_1:def 3;

            then

             A601: 1 <= k by FINSEQ_1: 1;

            k in ( Seg ( len v1)) by A579, FINSEQ_1:def 3;

            then k <= ( len v1) by FINSEQ_1: 1;

            then 1 <= ( len v1) by A601, XXREAL_0: 2;

            then

             A602: 1 in ( dom v1) by FINSEQ_3: 25;

            assume k <> 1;

            then k > 1 by A601, XXREAL_0: 1;

            hence contradiction by A579, A584, A602;

          end;

          ( min ( rng ( upper_volume (( chi (A,A)),D)))) in ( rng ( upper_volume (( chi (A,A)),D))) by XXREAL_2:def 7;

          then

          consider m such that

           A603: m in ( dom ( upper_volume (( chi (A,A)),D))) and

           A604: ( min ( rng ( upper_volume (( chi (A,A)),D)))) = (( upper_volume (( chi (A,A)),D)) . m) by PARTFUN1: 3;

          m in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by A603, FINSEQ_1:def 3;

          then

           A605: m in ( Seg ( len D)) by INTEGRA1:def 6;

          then m in ( dom D) by FINSEQ_1:def 3;

          then

           A606: ( min ( rng ( upper_volume (( chi (A,A)),D)))) = ( vol ( divset (D,m))) by A604, INTEGRA1: 20;

          

           A607: (v . m) = F(m) by A571, A573, A605

          .= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A606;

          

           A608: ( rng v) = ( rng v1) by A572, CLASSES1: 75;

          m in ( dom v) by A571, A605, FINSEQ_1:def 3;

          then ( min ( rng ( upper_volume (( chi (A,A)),D)))) in ( rng v) by A607, FUNCT_1:def 3;

          then

          consider m1 be Element of NAT such that

           A609: m1 in ( dom v1) and

           A610: ( min ( rng ( upper_volume (( chi (A,A)),D)))) = (v1 . m1) by A608, PARTFUN1: 3;

          (v1 . k) in ( rng v1) by A579, FUNCT_1:def 3;

          then

          consider k2 be Element of NAT such that

           A611: k2 in ( dom v) and

           A612: (v1 . k) = (v . k2) by A608, PARTFUN1: 3;

          

           A613: k2 in ( Seg ( len D)) by A571, A611, FINSEQ_1:def 3;

          then

           A614: k2 in ( dom D) by FINSEQ_1:def 3;

          k2 in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by A613, INTEGRA1:def 6;

          then

           A615: k2 in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

          (v1 . k) = F(k2) by A571, A611, A612

          .= ( vol ( divset (D,k2)));

          then (v1 . k) = (( upper_volume (( chi (A,A)),D)) . k2) by A614, INTEGRA1: 20;

          then (v1 . k) in ( rng ( upper_volume (( chi (A,A)),D))) by A615, FUNCT_1:def 3;

          then

           A616: (v1 . k) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by XXREAL_2:def 7;

          m1 >= 1 by A609, FINSEQ_3: 25;

          then (v1 . 1) <= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A579, A600, A609, A610, INTEGRA2: 2;

          hence thesis by A600, A616, XXREAL_0: 1;

        end;

        (H - h) <= ((H - h) + 1) by XREAL_1: 29;

        then

         A617: (p * (H - h)) <= (p * ((H - h) + 1)) by XREAL_1: 64;

        set sD = ( lower_sum (f,D));

        set s = ( lower_integral f);

        let m be Nat;

        reconsider mm = m as Element of NAT by ORDINAL1:def 12;

        reconsider D1 = (T . mm) as Division of A;

        

         A618: ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) <= (e / ((2 * p) * ((H - h) + 1))) by XXREAL_0: 17;

        assume

         A619: n <= m;

        then (( delta T) . m) < ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) by A582;

        then

         A620: ( delta D1) < ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) by Def2;

        (( delta T) . m) < ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) by A582, A619;

        then (( delta T) . m) < (e / ((2 * p) * ((H - h) + 1))) by A618, XXREAL_0: 2;

        then ((( delta T) . m) * ((2 * p) * ((H - h) + 1))) < e by A580, A565, XREAL_1: 79, XREAL_1: 129;

        then (((( delta T) . m) * (p * ((H - h) + 1))) * 2) < e;

        then

         A621: ((p * ((H - h) + 1)) * (( delta T) . m)) < (e / 2) by XREAL_1: 81;

        (T . mm) in ( divs A) by INTEGRA1:def 3;

        then

         A622: (T . mm) in ( dom ( lower_sum_set f)) by FUNCT_2:def 1;

        (( lower_sum (f,T)) . mm) = ( lower_sum (f,(T . mm))) by INTEGRA2:def 3;

        then (( lower_sum (f,T)) . m) = (( lower_sum_set f) . (T . m)) by INTEGRA1:def 11;

        then (( lower_sum (f,T)) . m) in ( rng ( lower_sum_set f)) by A622, FUNCT_1:def 3;

        then ( upper_bound ( rng ( lower_sum_set f))) >= (( lower_sum (f,T)) . m) by A566, SEQ_4:def 1;

        then ( lower_integral f) >= (( lower_sum (f,T)) . m) by INTEGRA1:def 15;

        then

         A623: (( lower_integral f) - (( lower_sum (f,T)) . m)) >= 0 by XREAL_1: 48;

         0 < (( delta T) . m) by A582, A619;

        then

         A624: ((p * (H - h)) * (( delta T) . m)) <= ((p * ((H - h) + 1)) * (( delta T) . m)) by A617, XREAL_1: 64;

        set sD1 = ( lower_sum (f,(T . mm)));

        consider D2 be Division of A such that

         A625: D <= D2 and D1 <= D2 and

         A626: ( rng D2) = (( rng D1) \/ ( rng D)) and 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D))) and 0 <= (( lower_sum (f,D2)) - ( lower_sum (f,D1))) by A5;

        set sD2 = ( lower_sum (f,D2));

        

         A627: ((sD - sD1) - (sD2 - sD1)) = (sD - sD2);

        ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) <= (v1 . k) by XXREAL_0: 17;

        then ( delta D1) < (v1 . k) by A620, XXREAL_0: 2;

        then ex D3 be Division of A st D <= D3 & D1 <= D3 & ( rng D3) = (( rng D1) \/ ( rng D)) & (( lower_sum (f,D3)) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A10, A599;

        then

         A628: (( lower_sum (f,D2)) - ( lower_sum (f,D1))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A626, Th6;

        (( lower_sum (f,D)) - ( lower_sum (f,D2))) <= 0 by A1, A625, INTEGRA1: 46, XREAL_1: 47;

        then

         A629: (sD - sD1) <= (sD2 - sD1) by A627, XREAL_1: 50;

        ( delta D1) = (( delta T) . m) by Def2;

        then (( lower_sum (f,D2)) - ( lower_sum (f,(T . mm)))) <= ((p * ((H - h) + 1)) * (( delta T) . m)) by A628, A624, XXREAL_0: 2;

        then (sD - sD1) <= ((p * ((H - h) + 1)) * (( delta T) . m)) by A629, XXREAL_0: 2;

        then (sD - sD1) < (e / 2) by A621, XXREAL_0: 2;

        then

         A630: ((sD - sD1) + (e / 2)) < ((e / 2) + (e / 2)) by XREAL_1: 6;

        ((s - sD1) + sD1) < (sD + (e / 2)) by A568, A583, XREAL_1: 19;

        then (s - sD1) < ((sD + (e / 2)) - sD1) by XREAL_1: 20;

        then (s - sD1) < e by A630, XXREAL_0: 2;

        then (( lower_integral f) - (( lower_sum (f,T)) . m)) < e by INTEGRA2:def 3;

        then |.(( lower_integral f) - (( lower_sum (f,T)) . m)).| < e by A623, ABSVALUE:def 1;

        then |.( - (( lower_integral f) - (( lower_sum (f,T)) . m))).| < e by COMPLEX1: 52;

        hence thesis;

      end;

      hence ( lower_sum (f,T)) is convergent by SEQ_2:def 6;

      hence thesis by A562, SEQ_2:def 7;

    end;

    theorem :: INTEGRA3:26

    (f | A) is bounded & ( delta T) is 0 -convergent non-zero & ( vol A) <> 0 implies ( upper_sum (f,T)) is convergent & ( lim ( upper_sum (f,T))) = ( upper_integral f)

    proof

      assume

       A1: (f | A) is bounded;

      then

       A2: for D, D1 holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & 0 <= (( upper_sum (f,D)) - ( upper_sum (f,D2))) & 0 <= (( upper_sum (f,D1)) - ( upper_sum (f,D2))) by Th22;

      

       A7: for D, D1 st ( delta D1) < ( min ( rng ( upper_volume (( chi (A,A)),D)))) holds ex D2 st D <= D2 & D1 <= D2 & ( rng D2) = (( rng D1) \/ ( rng D)) & (( upper_sum (f,D1)) - ( upper_sum (f,D2))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A1, Th23;

      assume

       A559: ( delta T) is 0 -convergent non-zero;

      then

       A560: ( delta T) is convergent by FDIFF_1:def 1;

      

       A561: ( lim ( delta T)) = 0 by A559, FDIFF_1:def 1;

      assume

       A562: ( vol A) <> 0 ;

      

       A563: ( delta T) is non-zero by A559;

      

       A564: for e st e > 0 holds ex n be Nat st for m be Nat st n <= m holds 0 < (( delta T) . m) & (( delta T) . m) < e

      proof

        let e;

        assume e > 0 ;

        then

        consider n be Nat such that

         A565: for m be Nat st n <= m holds |.((( delta T) . m) - 0 ).| < e by A560, A561, SEQ_2:def 7;

        take n;

        let m be Nat;

        reconsider mm = m as Element of NAT by ORDINAL1:def 12;

        assume n <= m;

        then |.((( delta T) . m) - 0 ).| < e by A565;

        then

         A566: ((( delta T) . m) + |.((( delta T) . m) - 0 ).|) < (e + |.((( delta T) . m) - 0 ).|) by ABSVALUE: 4, XREAL_1: 8;

        reconsider D = (T . mm) as Division of A;

        

         A567: (( delta T) . m) = ( delta (T . mm)) by Def2;

        ( delta (T . mm)) in ( rng ( upper_volume (( chi (A,A)),(T . mm)))) by XXREAL_2:def 8;

        then

        consider i such that

         A568: i in ( dom ( upper_volume (( chi (A,A)),(T . mm)))) and

         A569: ( delta (T . mm)) = (( upper_volume (( chi (A,A)),(T . mm))) . i) by PARTFUN1: 3;

        i in ( Seg ( len ( upper_volume (( chi (A,A)),(T . mm))))) by A568, FINSEQ_1:def 3;

        then i in ( Seg ( len D)) by INTEGRA1:def 6;

        then i in ( dom D) by FINSEQ_1:def 3;

        then

         A570: ( delta (T . mm)) = ( vol ( divset ((T . mm),i))) by A569, INTEGRA1: 20;

        (( delta T) . m) <> 0 by A563, SEQ_1: 5;

        hence thesis by A566, A567, A570, INTEGRA1: 9, XREAL_1: 6;

      end;

      

       A571: for e be Real st e > 0 holds ex n be Nat st for m be Nat st n <= m holds |.((( upper_sum (f,T)) . m) - ( upper_integral f)).| < e

      proof

        let e be Real;

        assume

         A572: e > 0 ;

        then

         A573: (e / 2) > 0 by XREAL_1: 139;

        reconsider e as Real;

        

         A574: ( rng ( upper_sum_set f)) is bounded_below by A1, INTEGRA2: 35;

        ( upper_integral f) = ( lower_bound ( rng ( upper_sum_set f))) by INTEGRA1:def 14;

        then

        consider y be Real such that

         A575: y in ( rng ( upper_sum_set f)) and

         A576: (( upper_integral f) + (e / 2)) > y by A573, A574, SEQ_4:def 2;

        ex D be Division of A st D in ( dom ( upper_sum_set f)) & y = (( upper_sum_set f) . D) & (D . 1) > ( lower_bound A)

        proof

          consider D3 be Element of ( divs A) such that

           A577: D3 in ( dom ( upper_sum_set f)) and

           A578: y = (( upper_sum_set f) . D3) by A575, PARTFUN1: 3;

          reconsider D3 as Division of A by INTEGRA1:def 3;

          

           A579: ( len D3) in ( Seg ( len D3)) by FINSEQ_1: 3;

          then 1 <= ( len D3) by FINSEQ_1: 1;

          then 1 in ( Seg ( len D3)) by FINSEQ_1: 1;

          then

           A580: 1 in ( dom D3) by FINSEQ_1:def 3;

          per cases ;

            suppose

             A581: (D3 . 1) <> ( lower_bound A);

            (D3 . 1) in A by A580, INTEGRA1: 6;

            then ( lower_bound A) <= (D3 . 1) by INTEGRA2: 1;

            then (D3 . 1) > ( lower_bound A) by A581, XXREAL_0: 1;

            hence thesis by A577, A578;

          end;

            suppose

             A582: (D3 . 1) = ( lower_bound A);

            ex D be Division of A st D in ( dom ( upper_sum_set f)) & y = (( upper_sum_set f) . D) & (D . 1) > ( lower_bound A)

            proof

              

               A583: (( upper_volume (f,D3)) . 1) = (( upper_bound ( rng (f | ( divset (D3,1))))) * ( vol ( divset (D3,1)))) by A580, INTEGRA1:def 6;

              ( vol A) >= 0 by INTEGRA1: 9;

              then

               A584: (( upper_bound A) - ( lower_bound A)) > 0 by A562, INTEGRA1:def 5;

              

               A585: y = ( upper_sum (f,D3)) by A578, INTEGRA1:def 10

              .= ( Sum ( upper_volume (f,D3))) by INTEGRA1:def 8

              .= ( Sum ((( upper_volume (f,D3)) | 1) ^ (( upper_volume (f,D3)) /^ 1))) by RFINSEQ: 8;

              

               A586: (D3 . ( len D3)) = ( upper_bound A) by INTEGRA1:def 2;

              ( len D3) in ( dom D3) by A579, FINSEQ_1:def 3;

              then

               A587: ( len D3) > 1 by A580, A582, A586, A584, SEQ_4: 137, XREAL_1: 47;

              then

              reconsider D = (D3 /^ 1) as increasing FinSequence of REAL by INTEGRA1: 34;

              

               A588: ( len D) = (( len D3) - 1) by A587, RFINSEQ:def 1;

              ( upper_bound A) > ( lower_bound A) by A584, XREAL_1: 47;

              then ( len D) <> 0 by A582, A588, INTEGRA1:def 2;

              then

              reconsider D as non empty increasing FinSequence of REAL ;

              

               A589: ( len D) in ( dom D) by FINSEQ_5: 6;

              (( len D) + 1) = ( len D3) by A588;

              then

               A590: (D . ( len D)) = ( upper_bound A) by A586, A587, A589, RFINSEQ:def 1;

              

               A591: ( len D) in ( Seg ( len D)) by FINSEQ_1: 3;

              (1 + 1) <= ( len D3) by A587, NAT_1: 13;

              then 2 in ( dom D3) by FINSEQ_3: 25;

              then

               A592: (D3 . 1) < (D3 . 2) by A580, SEQM_3:def 1;

              

               A593: ( rng D3) c= A by INTEGRA1:def 2;

              ( rng D) c= ( rng D3) by FINSEQ_5: 33;

              then ( rng D) c= A by A593;

              then

              reconsider D as Division of A by A590, INTEGRA1:def 2;

              

               A594: 1 in ( Seg 1) by FINSEQ_1: 1;

              

               A595: ( len D3) >= (1 + 1) by A587, NAT_1: 13;

              then

               A596: 2 <= ( len ( upper_volume (f,D3))) by INTEGRA1:def 6;

              1 <= ( len ( upper_volume (f,D3))) by A587, INTEGRA1:def 6;

              

              then

               A597: ( len ( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3)))))) = ((( len ( upper_volume (f,D3))) -' 2) + 1) by A596, FINSEQ_6: 118

              .= ((( len D3) -' 2) + 1) by INTEGRA1:def 6

              .= ((( len D3) - 2) + 1) by A595, XREAL_1: 233

              .= (( len D3) - 1);

              

               A598: for i be Nat st 1 <= i & i <= ( len ( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3)))))) holds (( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3))))) . i) = (( upper_volume (f,D)) . i)

              proof

                let i be Nat;

                assume that

                 A599: 1 <= i and

                 A600: i <= ( len ( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3))))));

                

                 A601: 1 <= (i + 1) by NAT_1: 12;

                (i + 1) <= ( len D3) by A597, A600, XREAL_1: 19;

                then

                 A602: (i + 1) in ( Seg ( len D3)) by A601, FINSEQ_1: 1;

                then

                 A603: (i + 1) in ( dom D3) by FINSEQ_1:def 3;

                

                 A604: ( divset (D3,(i + 1))) = ( divset (D,i))

                proof

                  

                   A605: (i + 1) in ( dom D3) by A602, FINSEQ_1:def 3;

                  

                   A606: 1 <> (i + 1) by A599, NAT_1: 13;

                  then

                   A607: ( upper_bound ( divset (D3,(i + 1)))) = (D3 . (i + 1)) by A605, INTEGRA1:def 4;

                  

                   A608: i in ( dom D) by A588, A597, A599, A600, FINSEQ_3: 25;

                  then

                   A609: (D . i) = (D3 . (i + 1)) by A587, RFINSEQ:def 1;

                  

                   A610: ( lower_bound ( divset (D3,(i + 1)))) = (D3 . ((i + 1) - 1)) by A606, A605, INTEGRA1:def 4;

                  per cases ;

                    suppose

                     A611: i = 1;

                    then

                     A612: ( upper_bound ( divset (D,i))) = (D . i) by A608, INTEGRA1:def 4;

                    

                     A613: ( lower_bound ( divset (D,i))) = ( lower_bound A) by A608, A611, INTEGRA1:def 4;

                    ( divset (D3,(i + 1))) = [.( lower_bound A), (D . i).] by A582, A607, A610, A609, A611, INTEGRA1: 4;

                    hence thesis by A613, A612, INTEGRA1: 4;

                  end;

                    suppose

                     A614: i <> 1;

                    then (i - 1) in ( dom D) by A608, INTEGRA1: 7;

                    

                    then

                     A615: (D . (i - 1)) = (D3 . ((i - 1) + 1)) by A587, RFINSEQ:def 1

                    .= (D3 . i);

                    

                     A616: ( upper_bound ( divset (D,i))) = (D . i) by A608, A614, INTEGRA1:def 4;

                    ( lower_bound ( divset (D,i))) = (D . (i - 1)) by A608, A614, INTEGRA1:def 4;

                    then ( divset (D3,(i + 1))) = [.( lower_bound ( divset (D,i))), ( upper_bound ( divset (D,i))).] by A607, A610, A609, A616, A615, INTEGRA1: 4;

                    hence thesis by INTEGRA1: 4;

                  end;

                end;

                i <= (( len ( upper_volume (f,D3))) - 1) by A597, A600, INTEGRA1:def 6;

                then

                 A617: i <= ((( len ( upper_volume (f,D3))) - 2) + 1);

                (( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3))))) . i) = (( upper_volume (f,D3)) . ((i + 2) - 1)) by A596, A599, A617, FINSEQ_6: 122

                .= (( upper_volume (f,D3)) . (i + 1));

                then

                 A618: (( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3))))) . i) = (( upper_bound ( rng (f | ( divset (D3,(i + 1)))))) * ( vol ( divset (D3,(i + 1))))) by A603, INTEGRA1:def 6;

                i in ( Seg ( len D)) by A588, A597, A599, A600, FINSEQ_1: 1;

                then i in ( dom D) by FINSEQ_1:def 3;

                hence thesis by A618, A604, INTEGRA1:def 6;

              end;

              

               A619: 1 <= ( len ( upper_volume (f,D3))) by A587, INTEGRA1:def 6;

              

               A620: ( len (( upper_volume (f,D3)) | 1)) = 1;

              1 in ( dom ( upper_volume (f,D3))) by A619, FINSEQ_3: 25;

              then ((( upper_volume (f,D3)) | 1) . 1) = (( upper_volume (f,D3)) . 1) by A594, RFINSEQ: 6;

              then

               A621: (( upper_volume (f,D3)) | 1) = <*(( upper_volume (f,D3)) . 1)*> by A620, FINSEQ_1: 40;

              

               A622: (2 -' 1) = (2 - 1) by XREAL_1: 233

              .= 1;

              1 <= ( len D) by A591, FINSEQ_1: 1;

              then 1 in ( dom D) by FINSEQ_3: 25;

              

              then

               A623: (D . 1) = (D3 . (1 + 1)) by A587, RFINSEQ:def 1

              .= (D3 . 2);

              D in ( divs A) by INTEGRA1:def 3;

              then

               A624: D in ( dom ( upper_sum_set f)) by FUNCT_2:def 1;

              ( len ( upper_volume (f,D3))) >= 2 by A595, INTEGRA1:def 6;

              then

               A625: ( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3))))) = (( upper_volume (f,D3)) /^ 1) by A622, FINSEQ_6: 117;

              ( len ( mid (( upper_volume (f,D3)),2,( len ( upper_volume (f,D3)))))) = ( len ( upper_volume (f,D))) by A588, A597, INTEGRA1:def 6;

              then

               A626: (( upper_volume (f,D3)) /^ 1) = ( upper_volume (f,D)) by A625, A598, FINSEQ_1: 14;

              ( vol ( divset (D3,1))) = (( upper_bound ( divset (D3,1))) - ( lower_bound ( divset (D3,1)))) by INTEGRA1:def 5

              .= (( upper_bound ( divset (D3,1))) - ( lower_bound A)) by A580, INTEGRA1:def 4

              .= ((D3 . 1) - ( lower_bound A)) by A580, INTEGRA1:def 4

              .= 0 by A582;

              

              then y = ( 0 + ( Sum ( upper_volume (f,D)))) by A585, A621, A583, A626, RVSUM_1: 76

              .= ( upper_sum (f,D)) by INTEGRA1:def 8;

              then y = (( upper_sum_set f) . D) by INTEGRA1:def 10;

              hence thesis by A582, A624, A623, A592;

            end;

            hence thesis;

          end;

        end;

        then

        consider D be Division of A such that D in ( dom ( upper_sum_set f)) and

         A627: y = (( upper_sum_set f) . D) and

         A628: (D . 1) > ( lower_bound A);

        deffunc F( Nat) = ( In (( vol ( divset (D,$1))), REAL ));

        set p = ( len D), H = ( upper_bound ( rng f)), h = ( lower_bound ( rng f));

        consider v be FinSequence of REAL such that

         A629: ( len v) = ( len D) & for j be Nat st j in ( dom v) holds (v . j) = F(j) from FINSEQ_2:sch 1;

        

         A630: (2 * p) > 0 by XREAL_1: 129;

        consider v1 be non-decreasing FinSequence of REAL such that

         A631: (v,v1) are_fiberwise_equipotent by INTEGRA2: 3;

        defpred P[ Nat] means $1 in ( dom v1) & (v1 . $1) > 0 ;

        

         A632: ( dom v) = ( Seg ( len D)) by A629, FINSEQ_1:def 3;

        

         A633: ex k be Nat st P[k]

        proof

          consider H be Function such that ( dom H) = ( dom v) and ( rng H) = ( dom v1) and H is one-to-one and

           A634: v = (v1 * H) by A631, CLASSES1: 77;

          consider k such that

           A635: k in ( dom D) and

           A636: ( vol ( divset (D,k))) > 0 by A562, Th2;

          

           A637: ( dom D) = ( Seg ( len D)) by FINSEQ_1:def 3;

          then (H . k) in ( dom v1) by A632, A634, A635, FUNCT_1: 11;

          then

          reconsider Hk = (H . k) as Element of NAT ;

          (v . k) = F(k) by A629, A632, A635, A637;

          then (v . k) > 0 by A636;

          then P[Hk] by A632, A634, A635, A637, FUNCT_1: 11, FUNCT_1: 12;

          hence thesis;

        end;

        consider k be Nat such that

         A638: P[k] & for n be Nat st P[n] holds k <= n from NAT_1:sch 5( A633);

        

         A639: (H - h) >= 0 by A1, Lm3, XREAL_1: 48;

        then

         A640: ((2 * p) * ((H - h) + 1)) > 0 by A630, XREAL_1: 129;

        ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) > 0

        proof

          per cases by XXREAL_0: 15;

            suppose ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) = (v1 . k);

            hence thesis by A638;

          end;

            suppose ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) = (e / ((2 * p) * ((H - h) + 1)));

            hence thesis by A572, A640, XREAL_1: 139;

          end;

        end;

        then

        consider n be Nat such that

         A641: for m be Nat st n <= m holds 0 < (( delta T) . m) & (( delta T) . m) < ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) by A564;

        take n;

        

         A642: y = ( upper_sum (f,D)) by A627, INTEGRA1:def 10;

        

         A643: (v1 . 1) > 0

        proof

          

           A644: for n1 be Element of NAT st n1 in ( dom D) holds ( vol ( divset (D,n1))) > 0

          proof

            let n1 be Element of NAT ;

            assume

             A645: n1 in ( dom D);

            then

             A646: 1 <= n1 by FINSEQ_3: 25;

            per cases by A646, XXREAL_0: 1;

              suppose

               A647: n1 = 1;

              then

               A648: ( upper_bound ( divset (D,n1))) = (D . n1) by A645, INTEGRA1:def 4;

              ( lower_bound ( divset (D,n1))) = ( lower_bound A) by A645, A647, INTEGRA1:def 4;

              then ( vol ( divset (D,n1))) = ((D . n1) - ( lower_bound A)) by A648, INTEGRA1:def 5;

              hence thesis by A628, A647, XREAL_1: 50;

            end;

              suppose

               A649: n1 > 1;

              then

               A650: ( upper_bound ( divset (D,n1))) = (D . n1) by A645, INTEGRA1:def 4;

              ( lower_bound ( divset (D,n1))) = (D . (n1 - 1)) by A645, A649, INTEGRA1:def 4;

              then

               A651: ( vol ( divset (D,n1))) = ((D . n1) - (D . (n1 - 1))) by A650, INTEGRA1:def 5;

              n1 < (n1 + 1) by XREAL_1: 29;

              then

               A652: (n1 - 1) < n1 by XREAL_1: 19;

              (n1 - 1) in ( dom D) by A645, A649, INTEGRA1: 7;

              then (D . (n1 - 1)) < (D . n1) by A645, A652, SEQM_3:def 1;

              hence thesis by A651, XREAL_1: 50;

            end;

          end;

          

           A653: k <= ( len v1) by A638, FINSEQ_3: 25;

          1 <= k by A638, FINSEQ_3: 25;

          then 1 <= ( len v1) by A653, XXREAL_0: 2;

          then 1 in ( dom v1) by FINSEQ_3: 25;

          then

           A654: (v1 . 1) in ( rng v1) by FUNCT_1:def 3;

          ( rng v) = ( rng v1) by A631, CLASSES1: 75;

          then

          consider n1 be Element of NAT such that

           A655: n1 in ( dom v) and

           A656: (v1 . 1) = (v . n1) by A654, PARTFUN1: 3;

          n1 in ( Seg ( len D)) by A629, A655, FINSEQ_1:def 3;

          then

           A657: n1 in ( dom D) by FINSEQ_1:def 3;

          (v1 . 1) = F(n1) by A629, A655, A656

          .= ( vol ( divset (D,n1)));

          hence thesis by A644, A657;

        end;

        

         A658: (v1 . k) = ( min ( rng ( upper_volume (( chi (A,A)),D))))

        proof

          

           A659: k = 1

          proof

            ( len v1) = ( len v) by A631, RFINSEQ: 3;

            then k in ( Seg ( len v)) by A638, FINSEQ_1:def 3;

            then

             A660: 1 <= k by FINSEQ_1: 1;

            k in ( Seg ( len v1)) by A638, FINSEQ_1:def 3;

            then k <= ( len v1) by FINSEQ_1: 1;

            then 1 <= ( len v1) by A660, XXREAL_0: 2;

            then

             A661: 1 in ( dom v1) by FINSEQ_3: 25;

            assume k <> 1;

            then k > 1 by A660, XXREAL_0: 1;

            hence contradiction by A638, A643, A661;

          end;

          

           A662: ( rng v) = ( rng v1) by A631, CLASSES1: 75;

          (v1 . k) in ( rng ( upper_volume (( chi (A,A)),D)))

          proof

            (v1 . k) in ( rng v) by A638, A662, FUNCT_1:def 3;

            then

            consider k2 be Element of NAT such that

             A663: k2 in ( dom v) and

             A664: (v1 . k) = (v . k2) by PARTFUN1: 3;

            

             A665: k2 in ( Seg ( len D)) by A629, A663, FINSEQ_1:def 3;

            then

             A666: k2 in ( dom D) by FINSEQ_1:def 3;

            k2 in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by A665, INTEGRA1:def 6;

            then

             A667: k2 in ( dom ( upper_volume (( chi (A,A)),D))) by FINSEQ_1:def 3;

            (v1 . k) = F(k2) by A629, A663, A664

            .= ( vol ( divset (D,k2)));

            then (v1 . k) = (( upper_volume (( chi (A,A)),D)) . k2) by A666, INTEGRA1: 20;

            hence thesis by A667, FUNCT_1:def 3;

          end;

          then

           A668: (v1 . k) >= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by XXREAL_2:def 7;

          ( min ( rng ( upper_volume (( chi (A,A)),D)))) in ( rng ( upper_volume (( chi (A,A)),D))) by XXREAL_2:def 7;

          then

          consider m such that

           A669: m in ( dom ( upper_volume (( chi (A,A)),D))) and

           A670: ( min ( rng ( upper_volume (( chi (A,A)),D)))) = (( upper_volume (( chi (A,A)),D)) . m) by PARTFUN1: 3;

          m in ( Seg ( len ( upper_volume (( chi (A,A)),D)))) by A669, FINSEQ_1:def 3;

          then

           A671: m in ( Seg ( len D)) by INTEGRA1:def 6;

          then m in ( dom D) by FINSEQ_1:def 3;

          then

           A672: ( min ( rng ( upper_volume (( chi (A,A)),D)))) = ( vol ( divset (D,m))) by A670, INTEGRA1: 20;

          

           A673: (v . m) = F(m) by A629, A632, A671

          .= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A672;

          m in ( dom v) by A629, A671, FINSEQ_1:def 3;

          then ( min ( rng ( upper_volume (( chi (A,A)),D)))) in ( rng v) by A673, FUNCT_1:def 3;

          then

          consider m1 be Element of NAT such that

           A674: m1 in ( dom v1) and

           A675: ( min ( rng ( upper_volume (( chi (A,A)),D)))) = (v1 . m1) by A662, PARTFUN1: 3;

          m1 >= 1 by A674, FINSEQ_3: 25;

          then (v1 . 1) <= ( min ( rng ( upper_volume (( chi (A,A)),D)))) by A638, A659, A674, A675, INTEGRA2: 2;

          hence thesis by A659, A668, XXREAL_0: 1;

        end;

        

         A676: ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) <= (v1 . k) by XXREAL_0: 17;

        set s = ( upper_integral f), sD = ( upper_sum (f,D));

        let m be Nat;

        reconsider mm = m as Element of NAT by ORDINAL1:def 12;

        reconsider D1 = (T . mm) as Division of A;

        

         A677: ( delta D1) = (( delta T) . m) by Def2;

        consider D2 be Division of A such that

         A678: D <= D2 and D1 <= D2 and

         A679: ( rng D2) = (( rng D1) \/ ( rng D)) and 0 <= (( upper_sum (f,D)) - ( upper_sum (f,D2))) and 0 <= (( upper_sum (f,D1)) - ( upper_sum (f,D2))) by A2;

        set sD1 = ( upper_sum (f,(T . mm))), sD2 = ( upper_sum (f,D2));

        ( upper_sum (f,D2)) <= ( upper_sum (f,D)) by A1, A678, INTEGRA1: 45;

        then

         A680: (sD1 - sD) <= (sD1 - sD2) by XREAL_1: 10;

        (((sD + sD1) - sD1) - s) < (e / 2) by A576, A642, XREAL_1: 19;

        then (((sD1 - s) + sD) - sD1) < (e / 2);

        then ((sD1 - s) + sD) < (sD1 + (e / 2)) by XREAL_1: 19;

        then

         A681: (sD1 - s) < ((sD1 + (e / 2)) - sD) by XREAL_1: 20;

        (T . mm) in ( divs A) by INTEGRA1:def 3;

        then

         A682: (T . m) in ( dom ( upper_sum_set f)) by FUNCT_2:def 1;

        (( upper_sum (f,T)) . m) = ( upper_sum (f,(T . mm))) by INTEGRA2:def 2;

        then (( upper_sum (f,T)) . m) = (( upper_sum_set f) . (T . m)) by INTEGRA1:def 10;

        then (( upper_sum (f,T)) . m) in ( rng ( upper_sum_set f)) by A682, FUNCT_1:def 3;

        then ( lower_bound ( rng ( upper_sum_set f))) <= (( upper_sum (f,T)) . m) by A574, SEQ_4:def 2;

        then ( upper_integral f) <= (( upper_sum (f,T)) . m) by INTEGRA1:def 14;

        then

         A683: ((( upper_sum (f,T)) . m) - ( upper_integral f)) >= 0 by XREAL_1: 48;

        (H - h) <= ((H - h) + 1) by XREAL_1: 29;

        then

         A684: (p * (H - h)) <= (p * ((H - h) + 1)) by XREAL_1: 64;

        

         A685: ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) <= (e / ((2 * p) * ((H - h) + 1))) by XXREAL_0: 17;

        assume

         A686: n <= m;

        then (( delta T) . m) < ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) by A641;

        then (( delta T) . m) < (e / ((2 * p) * ((H - h) + 1))) by A685, XXREAL_0: 2;

        then ((( delta T) . m) * ((2 * p) * ((H - h) + 1))) < e by A630, A639, XREAL_1: 79, XREAL_1: 129;

        then (((( delta T) . m) * (p * ((H - h) + 1))) * 2) < e;

        then

         A687: ((p * ((H - h) + 1)) * (( delta T) . m)) < (e / 2) by XREAL_1: 81;

        (( delta T) . m) < ( min ((v1 . k),(e / ((2 * p) * ((H - h) + 1))))) by A641, A686;

        then ( delta D1) < (v1 . k) by A677, A676, XXREAL_0: 2;

        then ex D3 be Division of A st D <= D3 & D1 <= D3 & ( rng D3) = (( rng D1) \/ ( rng D)) & (( upper_sum (f,D1)) - ( upper_sum (f,D3))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A7, A658;

        then

         A688: (( upper_sum (f,D1)) - ( upper_sum (f,D2))) <= ((( len D) * (( upper_bound ( rng f)) - ( lower_bound ( rng f)))) * ( delta D1)) by A679, Th6;

         0 < (( delta T) . m) by A641, A686;

        then ((p * (H - h)) * (( delta T) . m)) <= ((p * ((H - h) + 1)) * (( delta T) . m)) by A684, XREAL_1: 64;

        then (( upper_sum (f,(T . mm))) - ( upper_sum (f,D2))) <= ((p * ((H - h) + 1)) * (( delta T) . m)) by A677, A688, XXREAL_0: 2;

        then (sD1 - sD) <= ((p * ((H - h) + 1)) * (( delta T) . m)) by A680, XXREAL_0: 2;

        then (sD1 - sD) < (e / 2) by A687, XXREAL_0: 2;

        then ((sD1 - sD) + (e / 2)) < ((e / 2) + (e / 2)) by XREAL_1: 6;

        then (sD1 - s) < e by A681, XXREAL_0: 2;

        then ((( upper_sum (f,T)) . m) - ( upper_integral f)) < e by INTEGRA2:def 2;

        hence thesis by A683, ABSVALUE:def 1;

      end;

      hence ( upper_sum (f,T)) is convergent by SEQ_2:def 6;

      hence thesis by A571, SEQ_2:def 7;

    end;