let A be non empty closed_interval Subset of REAL; :: thesis: for Y being RealBanachSpace
for h being Function of A, the carrier of Y
for f being Function of A,REAL st h is bounded & h is integrable & f = ||.h.|| & f is integrable holds
||.(integral h).|| <= integral f
let Y be RealBanachSpace; :: thesis: for h being Function of A, the carrier of Y
for f being Function of A,REAL st h is bounded & h is integrable & f = ||.h.|| & f is integrable holds
||.(integral h).|| <= integral f
let h be Function of A, the carrier of Y; :: thesis: for f being Function of A,REAL st h is bounded & h is integrable & f = ||.h.|| & f is integrable holds
||.(integral h).|| <= integral f
let f be Function of A,REAL; :: thesis: ( h is bounded & h is integrable & f = ||.h.|| & f is integrable implies ||.(integral h).|| <= integral f )
assume A1: ( h is bounded & h is integrable & f = ||.h.|| & f is integrable ) ; :: thesis: ||.(integral h).|| <= integral f
then A2: f is bounded by Th1914;
consider T being DivSequence of A such that
A3: ( delta T is convergent & lim (delta T) = 0 ) by INTEGRA4:11;
set S = the middle_volume_Sequence of h,T;
A4: dom f = dom h by A1, NORMSP_0:def 3;
defpred S1[ Element of NAT , set ] means ex p being FinSequence of REAL st
( p = $2 & len p = len (T . $1) & ( for i being Nat st i in dom (T . $1) holds
( p . i in dom (h | (divset ((T . $1),i))) & ex z being Point of Y st
( z = (h | (divset ((T . $1),i))) . (p . i) & ( the middle_volume_Sequence of h,T . $1) . i = (vol (divset ((T . $1),i))) * z ) ) ) );
A5: for k being Element of NAT ex p being Element of REAL * st S1[k,p]
proof
let k be Element of NAT ; :: thesis: ex p being Element of REAL * st S1[k,p]
defpred S2[ Nat, set ] means ( $2 in dom (h | (divset ((T . k),$1))) & ex c being Point of Y st
( c = (h | (divset ((T . k),$1))) . $2 & ( the middle_volume_Sequence of h,T . k) . $1 = (vol (divset ((T . k),$1))) * c ) );
A6: Seg (len (T . k)) = dom (T . k) by FINSEQ_1:def 3;
A7: for i being Nat st i in Seg (len (T . k)) holds
ex x being Element of REAL st S2[i,x]
proof
let i be Nat; :: thesis: ( i in Seg (len (T . k)) implies ex x being Element of REAL st S2[i,x] )
assume i in Seg (len (T . k)) ; :: thesis: ex x being Element of REAL st S2[i,x]
then i in dom (T . k) by FINSEQ_1:def 3;
then consider c being Point of Y such that
A8: ( c in rng (h | (divset ((T . k),i))) & ( the middle_volume_Sequence of h,T . k) . i = (vol (divset ((T . k),i))) * c ) by INTEGR18:def 1;
consider x being object such that
A9: ( x in dom (h | (divset ((T . k),i))) & c = (h | (divset ((T . k),i))) . x ) by A8, FUNCT_1:def 3;
( x in dom h & x in divset ((T . k),i) ) by A9, RELAT_1:57;
then reconsider x = x as Element of REAL ;
take x ; :: thesis: S2[i,x]
thus S2[i,x] by A8, A9; :: thesis: verum
end;
consider p being FinSequence of REAL such that
A10: ( dom p = Seg (len (T . k)) & ( for i being Nat st i in Seg (len (T . k)) holds
S2[i,p . i] ) ) from FINSEQ_1:sch 5(A7);
 take p ; :: thesis: ( p is Element of REAL * & S1[k,p] )
len p = len (T . k) by A10, FINSEQ_1:def 3;
hence ( p is Element of REAL * & S1[k,p] ) by A10, A6, FINSEQ_1:def 11; :: thesis: verum
end;
consider F being sequence of (REAL *) such that
A11: for x being Element of NAT holds S1[x,F . x] from FUNCT_2:sch 3(A5);
 defpred S2[ Element of NAT , set ] means ex q being middle_volume of f,T . $1 st
( q = $2 & ( for i being Nat st i in dom (T . $1) holds
ex z being Element of REAL st
( (F . $1) . i in dom (f | (divset ((T . $1),i))) & z = (f | (divset ((T . $1),i))) . ((F . $1) . i) & q . i = (vol (divset ((T . $1),i))) * z ) ) );
A12: for k being Element of NAT ex y being Element of REAL * st S2[k,y]
proof
let k be Element of NAT ; :: thesis: ex y being Element of REAL * st S2[k,y]
defpred S3[ Nat, set ] means ex c being Element of REAL st
( (F . k) . $1 in dom (f | (divset ((T . k),$1))) & c = (f | (divset ((T . k),$1))) . ((F . k) . $1) & $2 = (vol (divset ((T . k),$1))) * c );
A13: Seg (len (T . k)) = dom (T . k) by FINSEQ_1:def 3;
A14: for i being Nat st i in Seg (len (T . k)) holds
ex x being Element of REAL st S3[i,x]
proof
let i be Nat; :: thesis: ( i in Seg (len (T . k)) implies ex x being Element of REAL st S3[i,x] )
assume i in Seg (len (T . k)) ; :: thesis: ex x being Element of REAL st S3[i,x]
then A15: i in dom (T . k) by FINSEQ_1:def 3;
consider p being FinSequence of REAL such that
A16: ( p = F . k & len p = len (T . k) & ( for i being Nat st i in dom (T . k) holds
( p . i in dom (h | (divset ((T . k),i))) & ex z being Point of Y st
( z = (h | (divset ((T . k),i))) . (p . i) & ( the middle_volume_Sequence of h,T . k) . i = (vol (divset ((T . k),i))) * z ) ) ) ) by A11;
p . i in dom (h | (divset ((T . k),i))) by A15, A16;
then A171: ( p . i in dom h & p . i in divset ((T . k),i) ) by RELAT_1:57;
( (vol (divset ((T . k),i))) * ((f | (divset ((T . k),i))) . (p . i)) in REAL & (f | (divset ((T . k),i))) . (p . i) in REAL ) by XREAL_0:def 1;
hence ex x being Element of REAL st S3[i,x] by A16, A171, A4, RELAT_1:57; :: thesis: verum
end;
consider q being FinSequence of REAL such that
A18: ( dom q = Seg (len (T . k)) & ( for i being Nat st i in Seg (len (T . k)) holds
S3[i,q . i] ) ) from FINSEQ_1:sch 5(A14);
 A19: len q = len (T . k) by A18, FINSEQ_1:def 3;
now :: thesis: for i being Nat st i in dom (T . k) holds
ex c being Element of REAL st
( c in rng (f | (divset ((T . k),i))) & q . i = c * (vol (divset ((T . k),i))) )
let i be Nat; :: thesis: ( i in dom (T . k) implies ex c being Element of REAL st
( c in rng (f | (divset ((T . k),i))) & q . i = c * (vol (divset ((T . k),i))) ) )
assume i in dom (T . k) ; :: thesis: ex c being Element of REAL st
( c in rng (f | (divset ((T . k),i))) & q . i = c * (vol (divset ((T . k),i))) )
then i in Seg (len (T . k)) by FINSEQ_1:def 3;
then ex c being Element of REAL st
( (F . k) . i in dom (f | (divset ((T . k),i))) & c = (f | (divset ((T . k),i))) . ((F . k) . i) & q . i = (vol (divset ((T . k),i))) * c ) by A18;
hence ex c being Element of REAL st
( c in rng (f | (divset ((T . k),i))) & q . i = c * (vol (divset ((T . k),i))) ) by FUNCT_1:3; :: thesis: verum
end;
then reconsider q = q as middle_volume of f,T . k by A19, INTEGR15:def 1;
q is Element of REAL * by FINSEQ_1:def 11;
hence ex y being Element of REAL * st S2[k,y] by A13, A18; :: thesis: verum
end;
consider Sf being sequence of (REAL *) such that
A21: for x being Element of NAT holds S2[x,Sf . x] from FUNCT_2:sch 3(A12);
now :: thesis: for k being Element of NAT holds Sf . k is middle_volume of f,T . k
let k be Element of NAT ; :: thesis: Sf . k is middle_volume of f,T . k
ex q being middle_volume of f,T . k st
( q = Sf . k & ( for i being Nat st i in dom (T . k) holds
ex z being Element of REAL st
( (F . k) . i in dom (f | (divset ((T . k),i))) & z = (f | (divset ((T . k),i))) . ((F . k) . i) & q . i = (vol (divset ((T . k),i))) * z ) ) ) by A21;
hence Sf . k is middle_volume of f,T . k ; :: thesis: verum
end;
then reconsider Sf = Sf as middle_volume_Sequence of f,T by INTEGR15:def 3;
A22: ( middle_sum (f,Sf) is convergent & lim (middle_sum (f,Sf)) = integral f ) by A1, A2, A3, INTEGR15:9;
A23: ( middle_sum (h, the middle_volume_Sequence of h,T) is convergent & lim (middle_sum (h, the middle_volume_Sequence of h,T)) = integral h ) by A1, A3, INTEGR18:def 6;
A24: for k being Element of NAT holds ||.((middle_sum (h, the middle_volume_Sequence of h,T)) . k).|| <= (middle_sum (f,Sf)) . k
proof
let k be Element of NAT ; :: thesis: ||.((middle_sum (h, the middle_volume_Sequence of h,T)) . k).|| <= (middle_sum (f,Sf)) . k
A25: (middle_sum (f,Sf)) . k = middle_sum (f,(Sf . k)) by INTEGR15:def 4;
A28: ex p being FinSequence of REAL st
( p = F . k & len p = len (T . k) & ( for i being Nat st i in dom (T . k) holds
( p . i in dom (h | (divset ((T . k),i))) & ex z being Point of Y st
( z = (h | (divset ((T . k),i))) . (p . i) & ( the middle_volume_Sequence of h,T . k) . i = (vol (divset ((T . k),i))) * z ) ) ) ) by A11;
A29: ex q being middle_volume of f,T . k st
( q = Sf . k & ( for i being Nat st i in dom (T . k) holds
ex z being Element of REAL st
( (F . k) . i in dom (f | (divset ((T . k),i))) & z = (f | (divset ((T . k),i))) . ((F . k) . i) & q . i = (vol (divset ((T . k),i))) * z ) ) ) by A21;
A30: len (Sf . k) = len (T . k) by INTEGR15:def 1;
A31: len ( the middle_volume_Sequence of h,T . k) = len (T . k) by INTEGR18:def 1;
now :: thesis: for i being Nat st i in dom ( the middle_volume_Sequence of h,T . k) holds
||.(( the middle_volume_Sequence of h,T . k) /. i).|| <= (Sf . k) . i
let i be Nat; :: thesis: ( i in dom ( the middle_volume_Sequence of h,T . k) implies ||.(( the middle_volume_Sequence of h,T . k) /. i).|| <= (Sf . k) . i )
assume A32: i in dom ( the middle_volume_Sequence of h,T . k) ; :: thesis: ||.(( the middle_volume_Sequence of h,T . k) /. i).|| <= (Sf . k) . i
then i in Seg (len (T . k)) by A31, FINSEQ_1:def 3;
then A34: i in dom (T . k) by FINSEQ_1:def 3;
then A36: (F . k) . i in dom (h | (divset ((T . k),i))) by A28;
consider z being Point of Y such that
A37: ( z = (h | (divset ((T . k),i))) . ((F . k) . i) & ( the middle_volume_Sequence of h,T . k) . i = (vol (divset ((T . k),i))) * z ) by A28, A34;
A38: ex w being Element of REAL st
( (F . k) . i in dom (f | (divset ((T . k),i))) & w = (f | (divset ((T . k),i))) . ((F . k) . i) & (Sf . k) . i = (vol (divset ((T . k),i))) * w ) by A29, A34;
( the middle_volume_Sequence of h,T . k) . i = ( the middle_volume_Sequence of h,T . k) /. i by A32, PARTFUN1:def 6;
then A41: ||.(( the middle_volume_Sequence of h,T . k) /. i).|| = |.(vol (divset ((T . k),i))).| * ||.z.|| by A37, NORMSP_1:def 1
.= (vol (divset ((T . k),i))) * ||.z.|| by INTEGRA1:9, ABSVALUE:def 1 ;
A42: ( dom (h | (divset ((T . k),i))) c= dom h & dom (f | (divset ((T . k),i))) c= dom f ) by RELAT_1:60;
A43: (h | (divset ((T . k),i))) . ((F . k) . i) = h . ((F . k) . i) by A36, FUNCT_1:47;
(f | (divset ((T . k),i))) . ((F . k) . i) = f . ((F . k) . i) by A38, FUNCT_1:47
.= ||.(h /. ((F . k) . i)).|| by A42, A1, A38, NORMSP_0:def 2 ;
hence ||.(( the middle_volume_Sequence of h,T . k) /. i).|| <= (Sf . k) . i by A41, A43, A38, A37, A42, A36, PARTFUN1:def 6; :: thesis: verum
end;
then ||.(middle_sum (h,( the middle_volume_Sequence of h,T . k))).|| <= Sum (Sf . k) by A30, A31, INTEGR20:10;
hence ||.((middle_sum (h, the middle_volume_Sequence of h,T)) . k).|| <= (middle_sum (f,Sf)) . k by A25, INTEGR18:def 4; :: thesis: verum
end;
A45: now :: thesis: for i being Nat holds ||.(middle_sum (h, the middle_volume_Sequence of h,T)).|| . i <= (middle_sum (f,Sf)) . i
let i be Nat; :: thesis: ||.(middle_sum (h, the middle_volume_Sequence of h,T)).|| . i <= (middle_sum (f,Sf)) . i
XX: i in NAT by ORDINAL1:def 12;
||.(middle_sum (h, the middle_volume_Sequence of h,T)).|| . i = ||.((middle_sum (h, the middle_volume_Sequence of h,T)) . i).|| by NORMSP_0:def 4;
hence ||.(middle_sum (h, the middle_volume_Sequence of h,T)).|| . i <= (middle_sum (f,Sf)) . i by A24, XX; :: thesis: verum
end;
||.(middle_sum (h, the middle_volume_Sequence of h,T)).|| is convergent by A1, A3, NORMSP_1:23;
then lim ||.(middle_sum (h, the middle_volume_Sequence of h,T)).|| <= lim (middle_sum (f,Sf)) by A45, A22, SEQ_2:18;
hence ||.(integral h).|| <= integral f by A22, A23, LOPBAN_1:41; :: thesis: verum