let X be RealNormSpace; :: thesis: for A being non empty closed_interval Subset of REAL
for f, g, h being Function of A, the carrier of X st h = f + g & f is integrable & g is integrable holds
( h is integrable & integral h = (integral f) + (integral g) )
let A be non empty closed_interval Subset of REAL; :: thesis: for f, g, h being Function of A, the carrier of X st h = f + g & f is integrable & g is integrable holds
( h is integrable & integral h = (integral f) + (integral g) )
let f, g, h be Function of A, the carrier of X; :: thesis: ( h = f + g & f is integrable & g is integrable implies ( h is integrable & integral h = (integral f) + (integral g) ) )
assume A1: ( h = f + g & f is integrable & g is integrable ) ; :: thesis: ( h is integrable & integral h = (integral f) + (integral g) )
A2: ( dom h = A & dom f = A & dom g = A ) by FUNCT_2:def 1;
A3: now :: thesis: for T being DivSequence of A
for S being middle_volume_Sequence of h,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (h,S) is convergent & lim (middle_sum (h,S)) = (integral f) + (integral g) )
let T be DivSequence of A; :: thesis: for S being middle_volume_Sequence of h,T st delta T is convergent & lim (delta T) = 0 holds
( middle_sum (h,S) is convergent & lim (middle_sum (h,S)) = (integral f) + (integral g) )
let S be middle_volume_Sequence of h,T; :: thesis: ( delta T is convergent & lim (delta T) = 0 implies ( middle_sum (h,S) is convergent & lim (middle_sum (h,S)) = (integral f) + (integral g) ) )
assume A4: ( delta T is convergent & lim (delta T) = 0 ) ; :: thesis: ( middle_sum (h,S) is convergent & lim (middle_sum (h,S)) = (integral f) + (integral g) )
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 X st
( z = (h | (divset ((T . $1),i))) . (p . i) & (S . $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 X st
( c = (h | (divset ((T . k),$1))) . $2 & (S . 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 X such that
A8: ( c in rng (h | (divset ((T . k),i))) & (S . k) . i = (vol (divset ((T . k),i))) * c ) by Def1;
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 Point of X 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 the carrier of X * st S2[k,y]
proof
let k be Element of NAT ; :: thesis: ex y being Element of the carrier of X * st S2[k,y]
defpred S3[ Nat, set ] means ex c being Point of X 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 the carrier of X st S3[i,x]
proof
let i be Nat; :: thesis: ( i in Seg (len (T . k)) implies ex x being Element of the carrier of X st S3[i,x] )
assume i in Seg (len (T . k)) ; :: thesis: ex x being Element of the carrier of X 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 X st
( z = (h | (divset ((T . k),i))) . (p . i) & (S . k) . i = (vol (divset ((T . k),i))) * z ) ) ) ) by A11;
p . i in dom (h | (divset ((T . k),i))) by A15, A16;
then A17: ( p . i in dom h & p . i in divset ((T . k),i) ) by RELAT_1:57;
then p . i in dom (f | (divset ((T . k),i))) by A2, RELAT_1:57;
then (f | (divset ((T . k),i))) . (p . i) in rng (f | (divset ((T . k),i))) by FUNCT_1:3;
then reconsider x = (f | (divset ((T . k),i))) . (p . i) as Element of the carrier of X ;
A18: (F . k) . i in dom (f | (divset ((T . k),i))) by A16, A17, A2, RELAT_1:57;
(vol (divset ((T . k),i))) * x is Element of the carrier of X ;
hence ex x being Element of the carrier of X st S3[i,x] by A16, A18; :: thesis: verum
end;
consider q being FinSequence of the carrier of X such that
A19: ( 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);
 A20: len q = len (T . k) by A19, FINSEQ_1:def 3;
now :: thesis: for i being Nat st i in dom (T . k) holds
ex c being Point of X st
( c in rng (f | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c )
let i be Nat; :: thesis: ( i in dom (T . k) implies ex c being Point of X st
( c in rng (f | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c ) )
assume i in dom (T . k) ; :: thesis: ex c being Point of X st
( c in rng (f | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c )
then i in Seg (len (T . k)) by FINSEQ_1:def 3;
then consider c being Point of X such that
A21: ( (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 A19;
thus ex c being Point of X st
( c in rng (f | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c ) by A21, FUNCT_1:3; :: thesis: verum
end;
then reconsider q = q as middle_volume of f,T . k by A20, Def1;
q is Element of the carrier of X * by FINSEQ_1:def 11;
hence ex y being Element of the carrier of X * st S2[k,y] by A13, A19; :: thesis: verum
end;
consider Sf being sequence of ( the carrier of X *) such that
A22: 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 Point of X 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 A22;
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 Def3;
defpred S3[ Element of NAT , set ] means ex q being middle_volume of g,T . $1 st
( q = $2 & ( for i being Nat st i in dom (T . $1) holds
ex z being Point of X st
( (F . $1) . i in dom (g | (divset ((T . $1),i))) & z = (g | (divset ((T . $1),i))) . ((F . $1) . i) & q . i = (vol (divset ((T . $1),i))) * z ) ) );
A23: for k being Element of NAT ex y being Element of the carrier of X * st S3[k,y]
proof
let k be Element of NAT ; :: thesis: ex y being Element of the carrier of X * st S3[k,y]
defpred S4[ Nat, set ] means ex c being Point of X st
( (F . k) . $1 in dom (g | (divset ((T . k),$1))) & c = (g | (divset ((T . k),$1))) . ((F . k) . $1) & $2 = (vol (divset ((T . k),$1))) * c );
A24: Seg (len (T . k)) = dom (T . k) by FINSEQ_1:def 3;
A25: for i being Nat st i in Seg (len (T . k)) holds
ex x being Element of the carrier of X st S4[i,x]
proof
let i be Nat; :: thesis: ( i in Seg (len (T . k)) implies ex x being Element of the carrier of X st S4[i,x] )
assume i in Seg (len (T . k)) ; :: thesis: ex x being Element of the carrier of X st S4[i,x]
then A26: i in dom (T . k) by FINSEQ_1:def 3;
consider p being FinSequence of REAL such that
A27: ( 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 X st
( z = (h | (divset ((T . k),i))) . (p . i) & (S . k) . i = (vol (divset ((T . k),i))) * z ) ) ) ) by A11;
p . i in dom (h | (divset ((T . k),i))) by A26, A27;
then A28: ( p . i in dom h & p . i in divset ((T . k),i) ) by RELAT_1:57;
then p . i in dom (g | (divset ((T . k),i))) by A2, RELAT_1:57;
then (g | (divset ((T . k),i))) . (p . i) in rng (g | (divset ((T . k),i))) by FUNCT_1:3;
then reconsider x = (g | (divset ((T . k),i))) . (p . i) as Element of the carrier of X ;
A29: (F . k) . i in dom (g | (divset ((T . k),i))) by A27, A28, A2, RELAT_1:57;
(vol (divset ((T . k),i))) * x is Element of the carrier of X ;
hence ex x being Element of the carrier of X st S4[i,x] by A27, A29; :: thesis: verum
end;
consider q being FinSequence of the carrier of X such that
A30: ( dom q = Seg (len (T . k)) & ( for i being Nat st i in Seg (len (T . k)) holds
S4[i,q . i] ) ) from FINSEQ_1:sch 5(A25);
 A31: len q = len (T . k) by A30, FINSEQ_1:def 3;
now :: thesis: for i being Nat st i in dom (T . k) holds
ex c being Point of X st
( c in rng (g | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c )
let i be Nat; :: thesis: ( i in dom (T . k) implies ex c being Point of X st
( c in rng (g | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c ) )
assume i in dom (T . k) ; :: thesis: ex c being Point of X st
( c in rng (g | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c )
then i in Seg (len (T . k)) by FINSEQ_1:def 3;
then consider c being Point of X such that
A32: ( (F . k) . i in dom (g | (divset ((T . k),i))) & c = (g | (divset ((T . k),i))) . ((F . k) . i) & q . i = (vol (divset ((T . k),i))) * c ) by A30;
thus ex c being Point of X st
( c in rng (g | (divset ((T . k),i))) & q . i = (vol (divset ((T . k),i))) * c ) by A32, FUNCT_1:3; :: thesis: verum
end;
then reconsider q = q as middle_volume of g,T . k by A31, Def1;
q is Element of the carrier of X * by FINSEQ_1:def 11;
hence ex y being Element of the carrier of X * st S3[k,y] by A24, A30; :: thesis: verum
end;
consider Sg being sequence of ( the carrier of X *) such that
A33: for x being Element of NAT holds S3[x,Sg . x] from FUNCT_2:sch 3(A23);
now :: thesis: for k being Element of NAT holds Sg . k is middle_volume of g,T . k
let k be Element of NAT ; :: thesis: Sg . k is middle_volume of g,T . k
ex q being middle_volume of g,T . k st
( q = Sg . k & ( for i being Nat st i in dom (T . k) holds
ex z being Point of X st
( (F . k) . i in dom (g | (divset ((T . k),i))) & z = (g | (divset ((T . k),i))) . ((F . k) . i) & q . i = (vol (divset ((T . k),i))) * z ) ) ) by A33;
hence Sg . k is middle_volume of g,T . k ; :: thesis: verum
end;
then reconsider Sg = Sg as middle_volume_Sequence of g,T by Def3;
A34: ( middle_sum (f,Sf) is convergent & lim (middle_sum (f,Sf)) = integral f ) by Def6, A1, A4;
A35: ( middle_sum (g,Sg) is convergent & lim (middle_sum (g,Sg)) = integral g ) by Def6, A1, A4;
A36: (middle_sum (f,Sf)) + (middle_sum (g,Sg)) = middle_sum (h,S)
proof
now :: thesis: for n being Nat holds ((middle_sum (f,Sf)) . n) + ((middle_sum (g,Sg)) . n) = (middle_sum (h,S)) . n
let n be Nat; :: thesis: ((middle_sum (f,Sf)) . n) + ((middle_sum (g,Sg)) . n) = (middle_sum (h,S)) . n
A37: n in NAT by ORDINAL1:def 12;
consider p being FinSequence of REAL such that
A38: ( p = F . n & len p = len (T . n) & ( for i being Nat st i in dom (T . n) holds
( p . i in dom (h | (divset ((T . n),i))) & ex z being Point of X st
( z = (h | (divset ((T . n),i))) . (p . i) & (S . n) . i = (vol (divset ((T . n),i))) * z ) ) ) ) by A11, A37;
consider q being middle_volume of f,T . n such that
A39: ( q = Sf . n & ( for i being Nat st i in dom (T . n) holds
ex z being Point of X st
( (F . n) . i in dom (f | (divset ((T . n),i))) & z = (f | (divset ((T . n),i))) . ((F . n) . i) & q . i = (vol (divset ((T . n),i))) * z ) ) ) by A22, A37;
consider r being middle_volume of g,T . n such that
A40: ( r = Sg . n & ( for i being Nat st i in dom (T . n) holds
ex z being Point of X st
( (F . n) . i in dom (g | (divset ((T . n),i))) & z = (g | (divset ((T . n),i))) . ((F . n) . i) & r . i = (vol (divset ((T . n),i))) * z ) ) ) by A33, A37;
A41: ( len (Sf . n) = len (T . n) & len (Sg . n) = len (T . n) & len (S . n) = len (T . n) ) by Def1;
A42: ( dom (Sf . n) = dom (T . n) & dom (Sg . n) = dom (T . n) & dom (S . n) = dom (T . n) ) by A41, FINSEQ_3:29;
now :: thesis: for i being Nat st 1 <= i & i <= len (S . n) holds
(S . n) /. i = ((Sf . n) /. i) + ((Sg . n) /. i)
let i be Nat; :: thesis: ( 1 <= i & i <= len (S . n) implies (S . n) /. i = ((Sf . n) /. i) + ((Sg . n) /. i) )
assume ( 1 <= i & i <= len (S . n) ) ; :: thesis: (S . n) /. i = ((Sf . n) /. i) + ((Sg . n) /. i)
then i in Seg (len (S . n)) by FINSEQ_1:1;
then A43: i in dom (S . n) by FINSEQ_1:def 3;
consider t being Point of X such that
A44: ( t = (h | (divset ((T . n),i))) . ((F . n) . i) & (S . n) . i = (vol (divset ((T . n),i))) * t ) by A43, A42, A38;
consider z being Point of X such that
A45: ( (F . n) . i in dom (f | (divset ((T . n),i))) & z = (f | (divset ((T . n),i))) . ((F . n) . i) & q . i = (vol (divset ((T . n),i))) * z ) by A39, A43, A42;
consider w being Point of X such that
A46: ( (F . n) . i in dom (g | (divset ((T . n),i))) & w = (g | (divset ((T . n),i))) . ((F . n) . i) & r . i = (vol (divset ((T . n),i))) * w ) by A40, A43, A42;
A47: (F . n) . i in divset ((T . n),i) by A46, RELAT_1:57;
A48: (F . n) . i in dom g by A46, RELAT_1:57;
A49: (F . n) . i in A by A46;
then A50: (F . n) . i in dom h by FUNCT_2:def 1;
A51: (F . n) . i in dom f by A49, FUNCT_2:def 1;
A52: f /. ((F . n) . i) = f . ((F . n) . i) by A51, PARTFUN1:def 6
.= z by A45, A47, FUNCT_1:49 ;
A53: g /. ((F . n) . i) = g . ((F . n) . i) by A48, PARTFUN1:def 6
.= w by A46, A47, FUNCT_1:49 ;
A54: t = (h | (divset ((T . n),i))) . ((F . n) . i) by A44
.= h . ((F . n) . i) by A47, FUNCT_1:49
.= h /. ((F . n) . i) by A50, PARTFUN1:def 6
.= (f /. ((F . n) . i)) + (g /. ((F . n) . i)) by A50, A1, VFUNCT_1:def 1
.= z + w by A52, A53 ;
A55: (vol (divset ((T . n),i))) * z = (Sf . n) . i by A45, A39
.= (Sf . n) /. i by A43, A42, PARTFUN1:def 6 ;
A56: (vol (divset ((T . n),i))) * w = (Sg . n) . i by A46, A40
.= (Sg . n) /. i by A43, A42, PARTFUN1:def 6 ;
thus (S . n) /. i = (S . n) . i by A43, PARTFUN1:def 6
.= (vol (divset ((T . n),i))) * t by A44
.= ((vol (divset ((T . n),i))) * z) + ((vol (divset ((T . n),i))) * w) by A54, RLVECT_1:def 5
.= ((Sf . n) /. i) + ((Sg . n) /. i) by A55, A56 ; :: thesis: verum
end;
then A57: (Sf . n) + (Sg . n) = S . n by A42, BINOM:def 1;
thus ((middle_sum (f,Sf)) . n) + ((middle_sum (g,Sg)) . n) = (middle_sum (f,(Sf . n))) + ((middle_sum (g,Sg)) . n) by Def4
.= (middle_sum (f,(Sf . n))) + (middle_sum (g,(Sg . n))) by Def4
.= (Sum (Sf . n)) + (middle_sum (g,(Sg . n)))
.= (Sum (Sf . n)) + (Sum (Sg . n))
.= Sum (S . n) by A57, A41, Th1
.= middle_sum (h,(S . n))
.= (middle_sum (h,S)) . n by Def4 ; :: thesis: verum
end;
hence (middle_sum (f,Sf)) + (middle_sum (g,Sg)) = middle_sum (h,S) by NORMSP_1:def 2; :: thesis: verum
end;
hence middle_sum (h,S) is convergent by A34, A35, NORMSP_1:19; :: thesis: lim (middle_sum (h,S)) = (integral f) + (integral g)
thus lim (middle_sum (h,S)) = (integral f) + (integral g) by A34, A35, A36, NORMSP_1:25; :: thesis: verum
end;
hence h is integrable ; :: thesis: integral h = (integral f) + (integral g)
hence integral h = (integral f) + (integral g) by Def6, A3; :: thesis: verum