:: INTEGRA1 semantic presentation
begin
registration
cluster closed_interval -> compact for Element of bool REAL;
coherence
for b1 being Subset of REAL st b1 is closed_interval holds
b1 is compact
proof
let IT be Subset of REAL; ::_thesis: ( IT is closed_interval implies IT is compact )
assume IT is closed_interval ; ::_thesis: IT is compact
then ex a, b being real number st IT = [.a,b.] by MEASURE5:def_3;
hence IT is compact by RCOMP_1:6; ::_thesis: verum
end;
end;
theorem :: INTEGRA1:1
canceled;
theorem :: INTEGRA1:2
canceled;
theorem Th3: :: INTEGRA1:3
for A being non empty closed_interval Subset of REAL holds
( A is bounded_below & A is bounded_above )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: ( A is bounded_below & A is bounded_above )
A is real-bounded by RCOMP_1:10;
hence ( A is bounded_below & A is bounded_above ) ; ::_thesis: verum
end;
registration
cluster non empty closed_interval -> real-bounded for Element of bool REAL;
coherence
for b1 being Subset of REAL st not b1 is empty & b1 is closed_interval holds
b1 is real-bounded
proof
let A be Subset of REAL; ::_thesis: ( not A is empty & A is closed_interval implies A is real-bounded )
assume ( not A is empty & A is closed_interval ) ; ::_thesis: A is real-bounded
hence ( A is bounded_below & A is bounded_above ) by Th3; :: according to XXREAL_2:def_11 ::_thesis: verum
end;
end;
theorem Th4: :: INTEGRA1:4
for A being non empty closed_interval Subset of REAL holds A = [.(lower_bound A),(upper_bound A).]
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: A = [.(lower_bound A),(upper_bound A).]
consider a, b being Real such that
A1: a <= b and
A2: A = [.a,b.] by MEASURE5:14;
A3: for y being real number st 0 < y holds
ex x being real number st
( x in A & b - y < x )
proof
let y be real number ; ::_thesis: ( 0 < y implies ex x being real number st
( x in A & b - y < x ) )
assume A4: 0 < y ; ::_thesis: ex x being real number st
( x in A & b - y < x )
take b ; ::_thesis: ( b in A & b - y < b )
b < b + y by A4, XREAL_1:29;
then b - y < (b + y) - y by XREAL_1:9;
hence ( b in A & b - y < b ) by A1, A2, XXREAL_1:1; ::_thesis: verum
end;
A5: for x being real number st x in A holds
x <= b
proof
let x be real number ; ::_thesis: ( x in A implies x <= b )
assume A6: x in A ; ::_thesis: x <= b
A = { y where y is Real : ( a <= y & y <= b ) } by A2, RCOMP_1:def_1;
then ex y being Real st
( x = y & a <= y & y <= b ) by A6;
hence x <= b ; ::_thesis: verum
end;
A7: for x being real number st x in A holds
a <= x
proof
let x be real number ; ::_thesis: ( x in A implies a <= x )
assume A8: x in A ; ::_thesis: a <= x
A = { y where y is Real : ( a <= y & y <= b ) } by A2, RCOMP_1:def_1;
then ex y being Real st
( x = y & a <= y & y <= b ) by A8;
hence a <= x ; ::_thesis: verum
end;
for y being real number st 0 < y holds
ex x being real number st
( x in A & x < a + y )
proof
let y be real number ; ::_thesis: ( 0 < y implies ex x being real number st
( x in A & x < a + y ) )
assume A9: 0 < y ; ::_thesis: ex x being real number st
( x in A & x < a + y )
take a ; ::_thesis: ( a in A & a < a + y )
thus ( a in A & a < a + y ) by A1, A2, A9, XREAL_1:29, XXREAL_1:1; ::_thesis: verum
end;
then a = lower_bound A by A7, SEQ_4:def_2;
hence A = [.(lower_bound A),(upper_bound A).] by A2, A5, A3, SEQ_4:def_1; ::_thesis: verum
end;
theorem Th5: :: INTEGRA1:5
for A being non empty closed_interval Subset of REAL
for a1, a2, b1, b2 being real number st A = [.a1,b1.] & A = [.a2,b2.] holds
( a1 = a2 & b1 = b2 )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for a1, a2, b1, b2 being real number st A = [.a1,b1.] & A = [.a2,b2.] holds
( a1 = a2 & b1 = b2 )
let a1, a2, b1, b2 be real number ; ::_thesis: ( A = [.a1,b1.] & A = [.a2,b2.] implies ( a1 = a2 & b1 = b2 ) )
assume that
A1: A = [.a1,b1.] and
A2: A = [.a2,b2.] ; ::_thesis: ( a1 = a2 & b1 = b2 )
a1 <= b1 by A1, XXREAL_1:29;
hence ( a1 = a2 & b1 = b2 ) by A1, A2, XXREAL_1:66; ::_thesis: verum
end;
begin
definition
canceled;
let A be non empty compact Subset of REAL;
mode Division of A -> non empty increasing FinSequence of REAL means :Def2: :: INTEGRA1:def 2
( rng it c= A & it . (len it) = upper_bound A );
existence
ex b1 being non empty increasing FinSequence of REAL st
( rng b1 c= A & b1 . (len b1) = upper_bound A )
proof
set a = upper_bound A;
reconsider p = (Seg 1) --> (upper_bound A) as Function of (Seg 1),REAL by FUNCOP_1:45;
A1: dom p = Seg 1 by FUNCOP_1:13;
rng p c= REAL ;
then reconsider p = p as FinSequence of REAL by FINSEQ_1:def_4;
A2: 1 in Seg 1 by FINSEQ_1:2, TARSKI:def_1;
for n, m being Element of NAT st n in dom p & m in dom p & n < m holds
p . n < p . m
proof
let n, m be Element of NAT ; ::_thesis: ( n in dom p & m in dom p & n < m implies p . n < p . m )
assume that
A3: n in dom p and
A4: m in dom p ; ::_thesis: ( not n < m or p . n < p . m )
n = 1 by A1, A3, FINSEQ_1:2, TARSKI:def_1;
hence ( not n < m or p . n < p . m ) by A1, A4, FINSEQ_1:2, TARSKI:def_1; ::_thesis: verum
end;
then A5: p is non empty increasing FinSequence of REAL by SEQM_3:def_1;
upper_bound A in A by RCOMP_1:14;
then for n being Nat st n in Seg 1 holds
p . n in A by FUNCOP_1:7;
then p is FinSequence of A by A1, FINSEQ_2:12;
then A6: rng p c= A by FINSEQ_1:def_4;
len p = 1 by A1, FINSEQ_1:def_3;
then p . (len p) = upper_bound A by A2, FUNCOP_1:7;
hence ex b1 being non empty increasing FinSequence of REAL st
( rng b1 c= A & b1 . (len b1) = upper_bound A ) by A6, A5; ::_thesis: verum
end;
end;
:: deftheorem INTEGRA1:def_1_:_
canceled;
:: deftheorem Def2 defines Division INTEGRA1:def_2_:_
for A being non empty compact Subset of REAL
for b2 being non empty increasing FinSequence of REAL holds
( b2 is Division of A iff ( rng b2 c= A & b2 . (len b2) = upper_bound A ) );
definition
let A be non empty compact Subset of REAL;
func divs A -> set means :Def3: :: INTEGRA1:def 3
for x1 being set holds
( x1 in it iff x1 is Division of A );
existence
ex b1 being set st
for x1 being set holds
( x1 in b1 iff x1 is Division of A )
proof
defpred S1[ set ] means $1 is Division of A;
consider R being set such that
A1: for x1 being set holds
( x1 in R iff ( x1 in bool [:NAT,REAL:] & S1[x1] ) ) from XBOOLE_0:sch_1();
take R ; ::_thesis: for x1 being set holds
( x1 in R iff x1 is Division of A )
let x1 be set ; ::_thesis: ( x1 in R iff x1 is Division of A )
thus ( x1 in R implies x1 is Division of A ) by A1; ::_thesis: ( x1 is Division of A implies x1 in R )
assume x1 is Division of A ; ::_thesis: x1 in R
then reconsider p = x1 as Division of A ;
p c= [:NAT,REAL:] ;
hence x1 in R by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being set st ( for x1 being set holds
( x1 in b1 iff x1 is Division of A ) ) & ( for x1 being set holds
( x1 in b2 iff x1 is Division of A ) ) holds
b1 = b2
proof
let D1, D2 be set ; ::_thesis: ( ( for x1 being set holds
( x1 in D1 iff x1 is Division of A ) ) & ( for x1 being set holds
( x1 in D2 iff x1 is Division of A ) ) implies D1 = D2 )
assume that
A2: for x1 being set holds
( x1 in D1 iff x1 is Division of A ) and
A3: for x1 being set holds
( x1 in D2 iff x1 is Division of A ) ; ::_thesis: D1 = D2
now__::_thesis:_for_x1_being_set_holds_
(_(_x1_in_D1_implies_x1_in_D2_)_&_(_x1_in_D2_implies_x1_in_D1_)_)
let x1 be set ; ::_thesis: ( ( x1 in D1 implies x1 in D2 ) & ( x1 in D2 implies x1 in D1 ) )
thus ( x1 in D1 implies x1 in D2 ) ::_thesis: ( x1 in D2 implies x1 in D1 )
proof
assume x1 in D1 ; ::_thesis: x1 in D2
then x1 is Division of A by A2;
hence x1 in D2 by A3; ::_thesis: verum
end;
assume x1 in D2 ; ::_thesis: x1 in D1
then x1 is Division of A by A3;
hence x1 in D1 by A2; ::_thesis: verum
end;
hence D1 = D2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines divs INTEGRA1:def_3_:_
for A being non empty compact Subset of REAL
for b2 being set holds
( b2 = divs A iff for x1 being set holds
( x1 in b2 iff x1 is Division of A ) );
registration
let A be non empty compact Subset of REAL;
cluster divs A -> non empty ;
coherence
not divs A is empty
proof
the Division of A in divs A by Def3;
hence not divs A is empty ; ::_thesis: verum
end;
end;
registration
let A be non empty compact Subset of REAL;
cluster -> Relation-like Function-like for Element of divs A;
coherence
for b1 being Element of divs A holds
( b1 is Function-like & b1 is Relation-like ) by Def3;
end;
registration
let A be non empty compact Subset of REAL;
cluster -> FinSequence-like real-valued for Element of divs A;
coherence
for b1 being Element of divs A holds
( b1 is real-valued & b1 is FinSequence-like ) by Def3;
end;
theorem Th6: :: INTEGRA1:6
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
D . i in A
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
D . i in A
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D holds
D . i in A
let D be Division of A; ::_thesis: ( i in dom D implies D . i in A )
assume i in dom D ; ::_thesis: D . i in A
then A1: D . i in rng D by FUNCT_1:def_3;
rng D c= A by Def2;
hence D . i in A by A1; ::_thesis: verum
end;
theorem Th7: :: INTEGRA1:7
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & i <> 1 holds
( i - 1 in dom D & D . (i - 1) in A & i - 1 in NAT )
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & i <> 1 holds
( i - 1 in dom D & D . (i - 1) in A & i - 1 in NAT )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D & i <> 1 holds
( i - 1 in dom D & D . (i - 1) in A & i - 1 in NAT )
let D be Division of A; ::_thesis: ( i in dom D & i <> 1 implies ( i - 1 in dom D & D . (i - 1) in A & i - 1 in NAT ) )
assume that
A1: i in dom D and
A2: i <> 1 ; ::_thesis: ( i - 1 in dom D & D . (i - 1) in A & i - 1 in NAT )
consider j being Nat such that
A3: dom D = Seg j by FINSEQ_1:def_2;
i <> 0 by A1, A3, FINSEQ_1:1;
then A4: not i is trivial by A2, NAT_2:def_1;
then consider l being Nat such that
A5: i = 2 + l by NAT_1:10, NAT_2:29;
reconsider l = l as Element of NAT by ORDINAL1:def_12;
i >= 2 by A4, NAT_2:29;
then A6: (1 + 1) - 1 <= i - 1 by XREAL_1:9;
i <= j by A1, A3, FINSEQ_1:1;
then A7: i - 1 <= j - 0 by XREAL_1:13;
A8: rng D c= A by Def2;
A9: l + 1 = i - (2 - 1) by A5;
then i - 1 in dom D by A3, A6, A7, FINSEQ_1:1;
then D . (i - 1) in rng D by FUNCT_1:def_3;
hence ( i - 1 in dom D & D . (i - 1) in A & i - 1 in NAT ) by A3, A6, A7, A9, A8, FINSEQ_1:1; ::_thesis: verum
end;
definition
let A be non empty closed_interval Subset of REAL;
let D be Division of A;
let i be Nat;
assume A1: i in dom D ;
func divset (D,i) -> non empty closed_interval Subset of REAL means :Def4: :: INTEGRA1:def 4
( lower_bound it = lower_bound A & upper_bound it = D . i ) if i = 1
otherwise ( lower_bound it = D . (i - 1) & upper_bound it = D . i );
existence
( ( i = 1 implies ex b1 being non empty closed_interval Subset of REAL st
( lower_bound b1 = lower_bound A & upper_bound b1 = D . i ) ) & ( not i = 1 implies ex b1 being non empty closed_interval Subset of REAL st
( lower_bound b1 = D . (i - 1) & upper_bound b1 = D . i ) ) )
proof
hereby ::_thesis: ( not i = 1 implies ex b1 being non empty closed_interval Subset of REAL st
( lower_bound b1 = D . (i - 1) & upper_bound b1 = D . i ) )
assume i = 1 ; ::_thesis: ex IT being non empty closed_interval Subset of REAL st
( lower_bound IT = lower_bound A & upper_bound IT = D . i )
thus ex IT being non empty closed_interval Subset of REAL st
( lower_bound IT = lower_bound A & upper_bound IT = D . i ) ::_thesis: verum
proof
consider I being Subset of REAL such that
A2: I = [.(lower_bound A),(D . i).] ;
D . i in A by A1, Th6;
then lower_bound A <= D . i by SEQ_4:def_2;
then A3: I is non empty closed_interval Subset of REAL by A2, MEASURE5:14;
then A4: I = [.(lower_bound I),(upper_bound I).] by Th4;
then A5: upper_bound I = D . i by A2, A3, Th5;
lower_bound I = lower_bound A by A2, A3, A4, Th5;
hence ex IT being non empty closed_interval Subset of REAL st
( lower_bound IT = lower_bound A & upper_bound IT = D . i ) by A3, A5; ::_thesis: verum
end;
end;
assume A6: i <> 1 ; ::_thesis: ex b1 being non empty closed_interval Subset of REAL st
( lower_bound b1 = D . (i - 1) & upper_bound b1 = D . i )
thus ex IT being non empty closed_interval Subset of REAL st
( lower_bound IT = D . (i - 1) & upper_bound IT = D . i ) ::_thesis: verum
proof
reconsider j = i - 1 as Element of NAT by A1, A6, Th7;
A7: i + (- 1) < i + (1 + (- 1)) by XREAL_1:6;
consider a1, b1 being Real such that
A8: a1 = D . (i - 1) and
A9: b1 = D . i ;
consider I being Subset of REAL such that
A10: I = [.a1,b1.] ;
i - 1 in dom D by A1, A6, Th7;
then D . j < D . i by A1, A7, SEQM_3:def_1;
then A11: I is non empty closed_interval Subset of REAL by A8, A9, A10, MEASURE5:14;
then A12: I = [.(lower_bound I),(upper_bound I).] by Th4;
then A13: upper_bound I = D . i by A9, A10, A11, Th5;
lower_bound I = D . (i - 1) by A8, A10, A11, A12, Th5;
hence ex IT being non empty closed_interval Subset of REAL st
( lower_bound IT = D . (i - 1) & upper_bound IT = D . i ) by A11, A13; ::_thesis: verum
end;
end;
uniqueness
for b1, b2 being non empty closed_interval Subset of REAL holds
( ( i = 1 & lower_bound b1 = lower_bound A & upper_bound b1 = D . i & lower_bound b2 = lower_bound A & upper_bound b2 = D . i implies b1 = b2 ) & ( not i = 1 & lower_bound b1 = D . (i - 1) & upper_bound b1 = D . i & lower_bound b2 = D . (i - 1) & upper_bound b2 = D . i implies b1 = b2 ) )
proof
let A1, A2 be non empty closed_interval Subset of REAL; ::_thesis: ( ( i = 1 & lower_bound A1 = lower_bound A & upper_bound A1 = D . i & lower_bound A2 = lower_bound A & upper_bound A2 = D . i implies A1 = A2 ) & ( not i = 1 & lower_bound A1 = D . (i - 1) & upper_bound A1 = D . i & lower_bound A2 = D . (i - 1) & upper_bound A2 = D . i implies A1 = A2 ) )
hereby ::_thesis: ( not i = 1 & lower_bound A1 = D . (i - 1) & upper_bound A1 = D . i & lower_bound A2 = D . (i - 1) & upper_bound A2 = D . i implies A1 = A2 )
consider b being Real such that
A14: b = D . i ;
assume that
i = 1 and
A15: lower_bound A1 = lower_bound A and
A16: upper_bound A1 = D . i and
A17: lower_bound A2 = lower_bound A and
A18: upper_bound A2 = D . i ; ::_thesis: A1 = A2
A1 = [.(lower_bound A),b.] by A15, A16, A14, Th4;
hence A1 = A2 by A17, A18, A14, Th4; ::_thesis: verum
end;
assume that
i <> 1 and
A19: lower_bound A1 = D . (i - 1) and
A20: upper_bound A1 = D . i and
A21: lower_bound A2 = D . (i - 1) and
A22: upper_bound A2 = D . i ; ::_thesis: A1 = A2
consider a, b being Real such that
A23: a = D . (i - 1) and
A24: b = D . i ;
A1 = [.a,b.] by A19, A20, A23, A24, Th4;
hence A1 = A2 by A21, A22, A23, A24, Th4; ::_thesis: verum
end;
correctness
consistency
for b1 being non empty closed_interval Subset of REAL holds verum;
;
end;
:: deftheorem Def4 defines divset INTEGRA1:def_4_:_
for A being non empty closed_interval Subset of REAL
for D being Division of A
for i being Nat st i in dom D holds
for b4 being non empty closed_interval Subset of REAL holds
( ( i = 1 implies ( b4 = divset (D,i) iff ( lower_bound b4 = lower_bound A & upper_bound b4 = D . i ) ) ) & ( not i = 1 implies ( b4 = divset (D,i) iff ( lower_bound b4 = D . (i - 1) & upper_bound b4 = D . i ) ) ) );
theorem Th8: :: INTEGRA1:8
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
divset (D,i) c= A
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
divset (D,i) c= A
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D holds
divset (D,i) c= A
let D be Division of A; ::_thesis: ( i in dom D implies divset (D,i) c= A )
assume A1: i in dom D ; ::_thesis: divset (D,i) c= A
now__::_thesis:_divset_(D,i)_c=_A
percases ( i = 1 or i <> 1 ) ;
supposeA2: i = 1 ; ::_thesis: divset (D,i) c= A
lower_bound A in A by RCOMP_1:14;
then A3: lower_bound A in [.(lower_bound A),(upper_bound A).] by Th4;
A4: lower_bound (divset (D,i)) = lower_bound A by A1, A2, Def4;
consider b being Real such that
A5: b = D . i ;
upper_bound (divset (D,i)) = b by A1, A2, A5, Def4;
then A6: divset (D,i) = [.(lower_bound A),b.] by A4, Th4;
b in A by A1, A5, Th6;
then b in [.(lower_bound A),(upper_bound A).] by Th4;
then [.(lower_bound A),b.] c= [.(lower_bound A),(upper_bound A).] by A3, XXREAL_2:def_12;
hence divset (D,i) c= A by A6, Th4; ::_thesis: verum
end;
supposeA7: i <> 1 ; ::_thesis: divset (D,i) c= A
set b = D . i;
D . i in A by A1, Th6;
then A8: D . i in [.(lower_bound A),(upper_bound A).] by Th4;
set a = D . (i - 1);
D . (i - 1) in A by A1, A7, Th7;
then D . (i - 1) in [.(lower_bound A),(upper_bound A).] by Th4;
then A9: [.(D . (i - 1)),(D . i).] c= [.(lower_bound A),(upper_bound A).] by A8, XXREAL_2:def_12;
A10: upper_bound (divset (D,i)) = D . i by A1, A7, Def4;
lower_bound (divset (D,i)) = D . (i - 1) by A1, A7, Def4;
then divset (D,i) = [.(D . (i - 1)),(D . i).] by A10, Th4;
hence divset (D,i) c= A by A9, Th4; ::_thesis: verum
end;
end;
end;
hence divset (D,i) c= A ; ::_thesis: verum
end;
definition
let A be Subset of REAL;
func vol A -> Real equals :: INTEGRA1:def 5
(upper_bound A) - (lower_bound A);
correctness
coherence
(upper_bound A) - (lower_bound A) is Real;
;
end;
:: deftheorem defines vol INTEGRA1:def_5_:_
for A being Subset of REAL holds vol A = (upper_bound A) - (lower_bound A);
theorem :: INTEGRA1:9
for A being non empty real-bounded Subset of REAL holds 0 <= vol A by SEQ_4:11, XREAL_1:48;
begin
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
let D be Division of A;
func upper_volume (f,D) -> FinSequence of REAL means :Def6: :: INTEGRA1:def 6
( len it = len D & ( for i being Nat st i in dom D holds
it . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = len D & ( for i being Nat st i in dom D holds
b1 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) )
proof
deffunc H1( Nat) -> Element of REAL = (upper_bound (rng (f | (divset (D,$1))))) * (vol (divset (D,$1)));
consider IT being FinSequence of REAL such that
A1: ( len IT = len D & ( for i being Nat st i in dom IT holds
IT . i = H1(i) ) ) from FINSEQ_2:sch_1();
take IT ; ::_thesis: ( len IT = len D & ( for i being Nat st i in dom D holds
IT . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) )
thus len IT = len D by A1; ::_thesis: for i being Nat st i in dom D holds
IT . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
let i be Nat; ::_thesis: ( i in dom D implies IT . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
assume i in dom D ; ::_thesis: IT . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
then i in dom IT by A1, FINSEQ_3:29;
hence IT . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence of REAL st len b1 = len D & ( for i being Nat st i in dom D holds
b1 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) & len b2 = len D & ( for i being Nat st i in dom D holds
b2 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) holds
b1 = b2
proof
let s1, s2 be FinSequence of REAL ; ::_thesis: ( len s1 = len D & ( for i being Nat st i in dom D holds
s1 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) & len s2 = len D & ( for i being Nat st i in dom D holds
s2 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) implies s1 = s2 )
assume that
A2: len s1 = len D and
A3: for i being Nat st i in dom D holds
s1 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) and
A4: len s2 = len D and
A5: for i being Nat st i in dom D holds
s2 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ; ::_thesis: s1 = s2
A6: dom s1 = dom D by A2, FINSEQ_3:29;
for i being Nat st i in dom s1 holds
s1 . i = s2 . i
proof
let i be Nat; ::_thesis: ( i in dom s1 implies s1 . i = s2 . i )
assume A7: i in dom s1 ; ::_thesis: s1 . i = s2 . i
then s1 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A3, A6;
hence s1 . i = s2 . i by A5, A6, A7; ::_thesis: verum
end;
hence s1 = s2 by A2, A4, FINSEQ_2:9; ::_thesis: verum
end;
func lower_volume (f,D) -> FinSequence of REAL means :Def7: :: INTEGRA1:def 7
( len it = len D & ( for i being Nat st i in dom D holds
it . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = len D & ( for i being Nat st i in dom D holds
b1 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) )
proof
deffunc H1( Nat) -> Element of REAL = (lower_bound (rng (f | (divset (D,$1))))) * (vol (divset (D,$1)));
consider IT being FinSequence of REAL such that
A8: ( len IT = len D & ( for i being Nat st i in dom IT holds
IT . i = H1(i) ) ) from FINSEQ_2:sch_1();
take IT ; ::_thesis: ( len IT = len D & ( for i being Nat st i in dom D holds
IT . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) )
thus len IT = len D by A8; ::_thesis: for i being Nat st i in dom D holds
IT . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
let i be Nat; ::_thesis: ( i in dom D implies IT . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
assume i in dom D ; ::_thesis: IT . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
then i in dom IT by A8, FINSEQ_3:29;
hence IT . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A8; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence of REAL st len b1 = len D & ( for i being Nat st i in dom D holds
b1 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) & len b2 = len D & ( for i being Nat st i in dom D holds
b2 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) holds
b1 = b2
proof
let s1, s2 be FinSequence of REAL ; ::_thesis: ( len s1 = len D & ( for i being Nat st i in dom D holds
s1 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) & len s2 = len D & ( for i being Nat st i in dom D holds
s2 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) implies s1 = s2 )
assume that
A9: len s1 = len D and
A10: for i being Nat st i in dom D holds
s1 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) and
A11: len s2 = len D and
A12: for i being Nat st i in dom D holds
s2 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ; ::_thesis: s1 = s2
A13: dom s1 = dom D by A9, FINSEQ_3:29;
for i being Nat st i in dom s1 holds
s1 . i = s2 . i
proof
let i be Nat; ::_thesis: ( i in dom s1 implies s1 . i = s2 . i )
assume A14: i in dom s1 ; ::_thesis: s1 . i = s2 . i
then s1 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A10, A13;
hence s1 . i = s2 . i by A12, A13, A14; ::_thesis: verum
end;
hence s1 = s2 by A9, A11, FINSEQ_2:9; ::_thesis: verum
end;
end;
:: deftheorem Def6 defines upper_volume INTEGRA1:def_6_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL
for D being Division of A
for b4 being FinSequence of REAL holds
( b4 = upper_volume (f,D) iff ( len b4 = len D & ( for i being Nat st i in dom D holds
b4 . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) ) );
:: deftheorem Def7 defines lower_volume INTEGRA1:def_7_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL
for D being Division of A
for b4 being FinSequence of REAL holds
( b4 = lower_volume (f,D) iff ( len b4 = len D & ( for i being Nat st i in dom D holds
b4 . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) ) ) );
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
let D be Division of A;
func upper_sum (f,D) -> Real equals :: INTEGRA1:def 8
Sum (upper_volume (f,D));
correctness
coherence
Sum (upper_volume (f,D)) is Real;
;
func lower_sum (f,D) -> Real equals :: INTEGRA1:def 9
Sum (lower_volume (f,D));
correctness
coherence
Sum (lower_volume (f,D)) is Real;
;
end;
:: deftheorem defines upper_sum INTEGRA1:def_8_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL
for D being Division of A holds upper_sum (f,D) = Sum (upper_volume (f,D));
:: deftheorem defines lower_sum INTEGRA1:def_9_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL
for D being Division of A holds lower_sum (f,D) = Sum (lower_volume (f,D));
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
set S = divs A;
func upper_sum_set f -> Function of (divs A),REAL means :Def10: :: INTEGRA1:def 10
for D being Division of A holds it . D = upper_sum (f,D);
existence
ex b1 being Function of (divs A),REAL st
for D being Division of A holds b1 . D = upper_sum (f,D)
proof
defpred S1[ Element of divs A, set ] means ex D being Division of A st
( D = $1 & $2 = upper_sum (f,D) );
A1: for x being Element of divs A ex y being Element of REAL st S1[x,y]
proof
let x be Element of divs A; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider x = x as Division of A by Def3;
take upper_sum (f,x) ; ::_thesis: S1[x, upper_sum (f,x)]
thus S1[x, upper_sum (f,x)] ; ::_thesis: verum
end;
consider g being Function of (divs A),REAL such that
A2: for x being Element of divs A holds S1[x,g . x] from FUNCT_2:sch_3(A1);
take g ; ::_thesis: for D being Division of A holds g . D = upper_sum (f,D)
let D be Division of A; ::_thesis: g . D = upper_sum (f,D)
reconsider D1 = D as Element of divs A by Def3;
S1[D1,g . D1] by A2;
hence g . D = upper_sum (f,D) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (divs A),REAL st ( for D being Division of A holds b1 . D = upper_sum (f,D) ) & ( for D being Division of A holds b2 . D = upper_sum (f,D) ) holds
b1 = b2
proof
let g1, g2 be Function of (divs A),REAL; ::_thesis: ( ( for D being Division of A holds g1 . D = upper_sum (f,D) ) & ( for D being Division of A holds g2 . D = upper_sum (f,D) ) implies g1 = g2 )
assume that
A3: for D being Division of A holds g1 . D = upper_sum (f,D) and
A4: for D being Division of A holds g2 . D = upper_sum (f,D) ; ::_thesis: g1 = g2
let a be Element of divs A; :: according to FUNCT_2:def_8 ::_thesis: g1 . a = g2 . a
reconsider d = a as Division of A by Def3;
thus g1 . a = upper_sum (f,d) by A3
.= g2 . a by A4 ; ::_thesis: verum
end;
func lower_sum_set f -> Function of (divs A),REAL means :Def11: :: INTEGRA1:def 11
for D being Division of A holds it . D = lower_sum (f,D);
existence
ex b1 being Function of (divs A),REAL st
for D being Division of A holds b1 . D = lower_sum (f,D)
proof
defpred S1[ Element of divs A, set ] means ex D being Division of A st
( D = $1 & $2 = lower_sum (f,D) );
A5: for x being Element of divs A ex y being Element of REAL st S1[x,y]
proof
let x be Element of divs A; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider x = x as Division of A by Def3;
take lower_sum (f,x) ; ::_thesis: S1[x, lower_sum (f,x)]
thus S1[x, lower_sum (f,x)] ; ::_thesis: verum
end;
consider g being Function of (divs A),REAL such that
A6: for x being Element of divs A holds S1[x,g . x] from FUNCT_2:sch_3(A5);
take g ; ::_thesis: for D being Division of A holds g . D = lower_sum (f,D)
let D be Division of A; ::_thesis: g . D = lower_sum (f,D)
reconsider D1 = D as Element of divs A by Def3;
S1[D1,g . D1] by A6;
hence g . D = lower_sum (f,D) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (divs A),REAL st ( for D being Division of A holds b1 . D = lower_sum (f,D) ) & ( for D being Division of A holds b2 . D = lower_sum (f,D) ) holds
b1 = b2
proof
let g1, g2 be Function of (divs A),REAL; ::_thesis: ( ( for D being Division of A holds g1 . D = lower_sum (f,D) ) & ( for D being Division of A holds g2 . D = lower_sum (f,D) ) implies g1 = g2 )
assume that
A7: for D being Division of A holds g1 . D = lower_sum (f,D) and
A8: for D being Division of A holds g2 . D = lower_sum (f,D) ; ::_thesis: g1 = g2
let a be Element of divs A; :: according to FUNCT_2:def_8 ::_thesis: g1 . a = g2 . a
reconsider d = a as Division of A by Def3;
thus g1 . a = lower_sum (f,d) by A7
.= g2 . a by A8 ; ::_thesis: verum
end;
end;
:: deftheorem Def10 defines upper_sum_set INTEGRA1:def_10_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL
for b3 being Function of (divs A),REAL holds
( b3 = upper_sum_set f iff for D being Division of A holds b3 . D = upper_sum (f,D) );
:: deftheorem Def11 defines lower_sum_set INTEGRA1:def_11_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL
for b3 being Function of (divs A),REAL holds
( b3 = lower_sum_set f iff for D being Division of A holds b3 . D = lower_sum (f,D) );
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
attrf is upper_integrable means :Def12: :: INTEGRA1:def 12
rng (upper_sum_set f) is bounded_below ;
attrf is lower_integrable means :Def13: :: INTEGRA1:def 13
rng (lower_sum_set f) is bounded_above ;
end;
:: deftheorem Def12 defines upper_integrable INTEGRA1:def_12_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL holds
( f is upper_integrable iff rng (upper_sum_set f) is bounded_below );
:: deftheorem Def13 defines lower_integrable INTEGRA1:def_13_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL holds
( f is lower_integrable iff rng (lower_sum_set f) is bounded_above );
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
func upper_integral f -> Real equals :: INTEGRA1:def 14
lower_bound (rng (upper_sum_set f));
correctness
coherence
lower_bound (rng (upper_sum_set f)) is Real;
;
end;
:: deftheorem defines upper_integral INTEGRA1:def_14_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL holds upper_integral f = lower_bound (rng (upper_sum_set f));
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
func lower_integral f -> Real equals :: INTEGRA1:def 15
upper_bound (rng (lower_sum_set f));
coherence
upper_bound (rng (lower_sum_set f)) is Real ;
end;
:: deftheorem defines lower_integral INTEGRA1:def_15_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL holds lower_integral f = upper_bound (rng (lower_sum_set f));
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
attrf is integrable means :Def16: :: INTEGRA1:def 16
( f is upper_integrable & f is lower_integrable & upper_integral f = lower_integral f );
end;
:: deftheorem Def16 defines integrable INTEGRA1:def_16_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL holds
( f is integrable iff ( f is upper_integrable & f is lower_integrable & upper_integral f = lower_integral f ) );
definition
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
func integral f -> Real equals :: INTEGRA1:def 17
upper_integral f;
coherence
upper_integral f is Real ;
end;
:: deftheorem defines integral INTEGRA1:def_17_:_
for A being non empty closed_interval Subset of REAL
for f being PartFunc of A,REAL holds integral f = upper_integral f;
begin
theorem Th10: :: INTEGRA1:10
for X being non empty set
for f, g being PartFunc of X,REAL holds rng (f + g) c= (rng f) ++ (rng g)
proof
let X be non empty set ; ::_thesis: for f, g being PartFunc of X,REAL holds rng (f + g) c= (rng f) ++ (rng g)
let f, g be PartFunc of X,REAL; ::_thesis: rng (f + g) c= (rng f) ++ (rng g)
for y being set st y in rng (f + g) holds
y in (rng f) ++ (rng g)
proof
let y be set ; ::_thesis: ( y in rng (f + g) implies y in (rng f) ++ (rng g) )
assume y in rng (f + g) ; ::_thesis: y in (rng f) ++ (rng g)
then consider x1 being set such that
A1: x1 in dom (f + g) and
A2: y = (f + g) . x1 by FUNCT_1:def_3;
A3: dom (f + g) = (dom f) /\ (dom g) by VALUED_1:def_1;
then x1 in dom f by A1, XBOOLE_0:def_4;
then A4: f . x1 in rng f by FUNCT_1:def_3;
x1 in dom g by A1, A3, XBOOLE_0:def_4;
then A5: g . x1 in rng g by FUNCT_1:def_3;
(f + g) . x1 = (f . x1) + (g . x1) by A1, VALUED_1:def_1;
hence y in (rng f) ++ (rng g) by A2, A4, A5, MEMBER_1:46; ::_thesis: verum
end;
hence rng (f + g) c= (rng f) ++ (rng g) by TARSKI:def_3; ::_thesis: verum
end;
theorem Th11: :: INTEGRA1:11
for X being non empty set
for f being PartFunc of X,REAL st f | X is bounded_below holds
rng f is bounded_below
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL st f | X is bounded_below holds
rng f is bounded_below
let f be PartFunc of X,REAL; ::_thesis: ( f | X is bounded_below implies rng f is bounded_below )
assume f | X is bounded_below ; ::_thesis: rng f is bounded_below
then consider a being real number such that
A1: for x1 being set st x1 in X /\ (dom f) holds
a <= f . x1 by RFUNCT_1:71;
A2: X /\ (dom f) = dom f by XBOOLE_1:28;
a is LowerBound of rng f
proof
let y be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not y in rng f or a <= y )
assume y in rng f ; ::_thesis: a <= y
then ex s being set st
( s in dom f & y = f . s ) by FUNCT_1:def_3;
hence a <= y by A1, A2; ::_thesis: verum
end;
hence rng f is bounded_below by XXREAL_2:def_9; ::_thesis: verum
end;
theorem :: INTEGRA1:12
for X being non empty set
for f being PartFunc of X,REAL st rng f is bounded_below holds
f | X is bounded_below
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL st rng f is bounded_below holds
f | X is bounded_below
let f be PartFunc of X,REAL; ::_thesis: ( rng f is bounded_below implies f | X is bounded_below )
assume rng f is bounded_below ; ::_thesis: f | X is bounded_below
then consider a being real number such that
A1: a is LowerBound of rng f by XXREAL_2:def_9;
for x1 being set st x1 in X /\ (dom f) holds
a <= f . x1
proof
let x1 be set ; ::_thesis: ( x1 in X /\ (dom f) implies a <= f . x1 )
A2: X /\ (dom f) = dom f by XBOOLE_1:28;
assume x1 in X /\ (dom f) ; ::_thesis: a <= f . x1
then f . x1 in rng f by A2, FUNCT_1:def_3;
hence a <= f . x1 by A1, XXREAL_2:def_2; ::_thesis: verum
end;
hence f | X is bounded_below by RFUNCT_1:71; ::_thesis: verum
end;
theorem Th13: :: INTEGRA1:13
for X being non empty set
for f being PartFunc of X,REAL st f | X is bounded_above holds
rng f is bounded_above
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL st f | X is bounded_above holds
rng f is bounded_above
let f be PartFunc of X,REAL; ::_thesis: ( f | X is bounded_above implies rng f is bounded_above )
assume f | X is bounded_above ; ::_thesis: rng f is bounded_above
then consider a being real number such that
A1: for x1 being set st x1 in X /\ (dom f) holds
f . x1 <= a by RFUNCT_1:70;
A2: X /\ (dom f) = dom f by XBOOLE_1:28;
a is UpperBound of rng f
proof
let y be ext-real number ; :: according to XXREAL_2:def_1 ::_thesis: ( not y in rng f or y <= a )
assume y in rng f ; ::_thesis: y <= a
then ex s being set st
( s in dom f & y = f . s ) by FUNCT_1:def_3;
hence y <= a by A1, A2; ::_thesis: verum
end;
hence rng f is bounded_above by XXREAL_2:def_10; ::_thesis: verum
end;
theorem :: INTEGRA1:14
for X being non empty set
for f being PartFunc of X,REAL st rng f is bounded_above holds
f | X is bounded_above
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL st rng f is bounded_above holds
f | X is bounded_above
let f be PartFunc of X,REAL; ::_thesis: ( rng f is bounded_above implies f | X is bounded_above )
assume rng f is bounded_above ; ::_thesis: f | X is bounded_above
then consider a being real number such that
A1: a is UpperBound of rng f by XXREAL_2:def_10;
for x1 being set st x1 in X /\ (dom f) holds
f . x1 <= a
proof
let x1 be set ; ::_thesis: ( x1 in X /\ (dom f) implies f . x1 <= a )
A2: X /\ (dom f) = dom f by XBOOLE_1:28;
assume x1 in X /\ (dom f) ; ::_thesis: f . x1 <= a
then f . x1 in rng f by A2, FUNCT_1:def_3;
hence f . x1 <= a by A1, XXREAL_2:def_1; ::_thesis: verum
end;
hence f | X is bounded_above by RFUNCT_1:70; ::_thesis: verum
end;
theorem :: INTEGRA1:15
for X being non empty set
for f being PartFunc of X,REAL st f | X is bounded holds
rng f is real-bounded
proof
let X be non empty set ; ::_thesis: for f being PartFunc of X,REAL st f | X is bounded holds
rng f is real-bounded
let f be PartFunc of X,REAL; ::_thesis: ( f | X is bounded implies rng f is real-bounded )
assume A1: f | X is bounded ; ::_thesis: rng f is real-bounded
then A2: rng f is bounded_above by Th13;
rng f is bounded_below by A1, Th11;
hence rng f is real-bounded by A2; ::_thesis: verum
end;
begin
theorem Th16: :: INTEGRA1:16
for A being non empty set holds (chi (A,A)) | A is V8()
proof
let A be non empty set ; ::_thesis: (chi (A,A)) | A is V8()
for x being Element of A st x in A /\ (dom (chi (A,A))) holds
(chi (A,A)) /. x = 1
proof
let x be Element of A; ::_thesis: ( x in A /\ (dom (chi (A,A))) implies (chi (A,A)) /. x = 1 )
assume x in A /\ (dom (chi (A,A))) ; ::_thesis: (chi (A,A)) /. x = 1
then A1: x in dom (chi (A,A)) by XBOOLE_0:def_4;
(chi (A,A)) . x = 1 by FUNCT_3:def_3;
hence (chi (A,A)) /. x = 1 by A1, PARTFUN1:def_6; ::_thesis: verum
end;
hence (chi (A,A)) | A is V8() by PARTFUN2:35; ::_thesis: verum
end;
theorem Th17: :: INTEGRA1:17
for X being non empty set
for A being non empty Subset of X holds rng (chi (A,A)) = {1}
proof
let X be non empty set ; ::_thesis: for A being non empty Subset of X holds rng (chi (A,A)) = {1}
let A be non empty Subset of X; ::_thesis: rng (chi (A,A)) = {1}
A1: (chi (A,A)) | A is V8() by Th16;
dom (chi (A,A)) = A by FUNCT_3:def_3;
then A2: A = A /\ (dom (chi (A,A))) ;
A3: dom (chi (A,A)) = A by FUNCT_3:def_3;
ex x being Element of X st
( x in dom (chi (A,A)) & (chi (A,A)) . x = 1 )
proof
consider x being Element of X such that
A4: x in dom (chi (A,A)) by A3, SUBSET_1:4;
take x ; ::_thesis: ( x in dom (chi (A,A)) & (chi (A,A)) . x = 1 )
thus ( x in dom (chi (A,A)) & (chi (A,A)) . x = 1 ) by A4, FUNCT_3:def_3; ::_thesis: verum
end;
then A5: 1 in rng (chi (A,A)) by FUNCT_1:def_3;
A meets dom (chi (A,A)) by A2, XBOOLE_0:def_7;
then ex y being Element of REAL st rng ((chi (A,A)) | A) = {y} by A1, PARTFUN2:37;
hence rng (chi (A,A)) = {1} by A5, TARSKI:def_1; ::_thesis: verum
end;
theorem Th18: :: INTEGRA1:18
for X being non empty set
for A being non empty Subset of X
for B being set st B meets dom (chi (A,A)) holds
rng ((chi (A,A)) | B) = {1}
proof
let X be non empty set ; ::_thesis: for A being non empty Subset of X
for B being set st B meets dom (chi (A,A)) holds
rng ((chi (A,A)) | B) = {1}
let A be non empty Subset of X; ::_thesis: for B being set st B meets dom (chi (A,A)) holds
rng ((chi (A,A)) | B) = {1}
let B be set ; ::_thesis: ( B meets dom (chi (A,A)) implies rng ((chi (A,A)) | B) = {1} )
A1: dom ((chi (A,A)) | B) = B /\ (dom (chi (A,A))) by RELAT_1:61;
rng ((chi (A,A)) | B) c= rng (chi (A,A)) by RELAT_1:70;
then A2: rng ((chi (A,A)) | B) c= {1} by Th17;
assume B /\ (dom (chi (A,A))) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: rng ((chi (A,A)) | B) = {1}
then rng ((chi (A,A)) | B) <> {} by A1, RELAT_1:42;
hence rng ((chi (A,A)) | B) = {1} by A2, ZFMISC_1:33; ::_thesis: verum
end;
theorem Th19: :: INTEGRA1:19
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
vol (divset (D,i)) = (lower_volume ((chi (A,A)),D)) . i
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
vol (divset (D,i)) = (lower_volume ((chi (A,A)),D)) . i
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D holds
vol (divset (D,i)) = (lower_volume ((chi (A,A)),D)) . i
let D be Division of A; ::_thesis: ( i in dom D implies vol (divset (D,i)) = (lower_volume ((chi (A,A)),D)) . i )
A1: dom (chi (A,A)) = A by FUNCT_3:def_3;
assume A2: i in dom D ; ::_thesis: vol (divset (D,i)) = (lower_volume ((chi (A,A)),D)) . i
then A3: (lower_volume ((chi (A,A)),D)) . i = (lower_bound (rng ((chi (A,A)) | (divset (D,i))))) * (vol (divset (D,i))) by Def7;
divset (D,i) c= A by A2, Th8;
then divset (D,i) c= (divset (D,i)) /\ (dom (chi (A,A))) by A1, XBOOLE_1:19;
then (divset (D,i)) /\ (dom (chi (A,A))) <> {} ;
then divset (D,i) meets dom (chi (A,A)) by XBOOLE_0:def_7;
then A4: rng ((chi (A,A)) | (divset (D,i))) = {1} by Th18;
A5: rng (chi (A,A)) = {1} by Th17;
then lower_bound (rng (chi (A,A))) = 1 by SEQ_4:9;
hence vol (divset (D,i)) = (lower_volume ((chi (A,A)),D)) . i by A3, A5, A4; ::_thesis: verum
end;
theorem Th20: :: INTEGRA1:20
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
vol (divset (D,i)) = (upper_volume ((chi (A,A)),D)) . i
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
vol (divset (D,i)) = (upper_volume ((chi (A,A)),D)) . i
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D holds
vol (divset (D,i)) = (upper_volume ((chi (A,A)),D)) . i
let D be Division of A; ::_thesis: ( i in dom D implies vol (divset (D,i)) = (upper_volume ((chi (A,A)),D)) . i )
A1: dom (chi (A,A)) = A by FUNCT_3:def_3;
assume A2: i in dom D ; ::_thesis: vol (divset (D,i)) = (upper_volume ((chi (A,A)),D)) . i
then A3: (upper_volume ((chi (A,A)),D)) . i = (upper_bound (rng ((chi (A,A)) | (divset (D,i))))) * (vol (divset (D,i))) by Def6;
divset (D,i) c= A by A2, Th8;
then divset (D,i) c= (divset (D,i)) /\ (dom (chi (A,A))) by A1, XBOOLE_1:19;
then (divset (D,i)) /\ (dom (chi (A,A))) <> {} ;
then divset (D,i) meets dom (chi (A,A)) by XBOOLE_0:def_7;
then A4: rng ((chi (A,A)) | (divset (D,i))) = {1} by Th18;
A5: rng (chi (A,A)) = {1} by Th17;
then upper_bound (rng (chi (A,A))) = 1 by SEQ_4:9;
hence vol (divset (D,i)) = (upper_volume ((chi (A,A)),D)) . i by A3, A5, A4; ::_thesis: verum
end;
theorem :: INTEGRA1:21
for F, G, H being FinSequence of REAL st len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) + (G /. k) ) holds
Sum H = (Sum F) + (Sum G)
proof
let F, G, H be FinSequence of REAL ; ::_thesis: ( len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) + (G /. k) ) implies Sum H = (Sum F) + (Sum G) )
assume that
A1: len F = len G and
A2: len F = len H and
A3: for k being Element of NAT st k in dom F holds
H . k = (F /. k) + (G /. k) ; ::_thesis: Sum H = (Sum F) + (Sum G)
A4: F is Element of (len F) -tuples_on REAL by FINSEQ_2:92;
A5: G is Element of (len F) -tuples_on REAL by A1, FINSEQ_2:92;
then F + G is Element of (len F) -tuples_on REAL by A4, FINSEQ_2:120;
then A6: len H = len (F + G) by A2, CARD_1:def_7;
then A7: dom H = Seg (len (F + G)) by FINSEQ_1:def_3;
A8: for k being Element of NAT st k in dom F holds
H . k = (F . k) + (G . k)
proof
let k be Element of NAT ; ::_thesis: ( k in dom F implies H . k = (F . k) + (G . k) )
assume A9: k in dom F ; ::_thesis: H . k = (F . k) + (G . k)
then k in Seg (len G) by A1, FINSEQ_1:def_3;
then k in dom G by FINSEQ_1:def_3;
then A10: G /. k = G . k by PARTFUN1:def_6;
F /. k = F . k by A9, PARTFUN1:def_6;
hence H . k = (F . k) + (G . k) by A3, A9, A10; ::_thesis: verum
end;
for k being Nat st k in dom H holds
H . k = (F + G) . k
proof
let k be Nat; ::_thesis: ( k in dom H implies H . k = (F + G) . k )
assume A11: k in dom H ; ::_thesis: H . k = (F + G) . k
then k in dom F by A2, A6, A7, FINSEQ_1:def_3;
then A12: H . k = (F . k) + (G . k) by A8;
k in dom (F + G) by A7, A11, FINSEQ_1:def_3;
hence H . k = (F + G) . k by A12, VALUED_1:def_1; ::_thesis: verum
end;
then Sum H = Sum (F + G) by A6, FINSEQ_2:9
.= (Sum F) + (Sum G) by A4, A5, RVSUM_1:89 ;
hence Sum H = (Sum F) + (Sum G) ; ::_thesis: verum
end;
theorem Th22: :: INTEGRA1:22
for F, G, H being FinSequence of REAL st len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)
proof
let F, G, H be FinSequence of REAL ; ::_thesis: ( len F = len G & len F = len H & ( for k being Element of NAT st k in dom F holds
H . k = (F /. k) - (G /. k) ) implies Sum H = (Sum F) - (Sum G) )
assume that
A1: len F = len G and
A2: len F = len H and
A3: for k being Element of NAT st k in dom F holds
H . k = (F /. k) - (G /. k) ; ::_thesis: Sum H = (Sum F) - (Sum G)
A4: F is Element of (len F) -tuples_on REAL by FINSEQ_2:92;
A5: G is Element of (len F) -tuples_on REAL by A1, FINSEQ_2:92;
then A6: F - G is Element of (len F) -tuples_on REAL by A4, FINSEQ_2:120;
then A7: len H = len (F - G) by A2, CARD_1:def_7;
then A8: dom H = Seg (len (F - G)) by FINSEQ_1:def_3;
A9: for k being Element of NAT st k in dom F holds
H . k = (F . k) - (G . k)
proof
let k be Element of NAT ; ::_thesis: ( k in dom F implies H . k = (F . k) - (G . k) )
assume A10: k in dom F ; ::_thesis: H . k = (F . k) - (G . k)
then k in Seg (len G) by A1, FINSEQ_1:def_3;
then k in dom G by FINSEQ_1:def_3;
then A11: G /. k = G . k by PARTFUN1:def_6;
F /. k = F . k by A10, PARTFUN1:def_6;
hence H . k = (F . k) - (G . k) by A3, A10, A11; ::_thesis: verum
end;
for k being Nat st k in dom H holds
H . k = (F - G) . k
proof
let k be Nat; ::_thesis: ( k in dom H implies H . k = (F - G) . k )
assume A12: k in dom H ; ::_thesis: H . k = (F - G) . k
then k in Seg (len F) by A6, A8, CARD_1:def_7;
then k in dom F by FINSEQ_1:def_3;
then A13: H . k = (F . k) - (G . k) by A9;
k in dom (F - G) by A8, A12, FINSEQ_1:def_3;
hence H . k = (F - G) . k by A13, VALUED_1:13; ::_thesis: verum
end;
then Sum H = Sum (F - G) by A7, FINSEQ_2:9
.= (Sum F) - (Sum G) by A4, A5, RVSUM_1:90 ;
hence Sum H = (Sum F) - (Sum G) ; ::_thesis: verum
end;
theorem Th23: :: INTEGRA1:23
for A being non empty closed_interval Subset of REAL
for D being Division of A holds Sum (lower_volume ((chi (A,A)),D)) = vol A
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A holds Sum (lower_volume ((chi (A,A)),D)) = vol A
let D be Division of A; ::_thesis: Sum (lower_volume ((chi (A,A)),D)) = vol A
deffunc H1( Nat) -> Real = vol (divset (D,$1));
consider p being FinSequence of REAL such that
A1: ( len p = len D & ( for i being Nat st i in dom p holds
p . i = H1(i) ) ) from FINSEQ_2:sch_1();
A2: dom p = Seg (len D) by A1, FINSEQ_1:def_3;
A3: for i being Element of NAT st i in Seg (len D) holds
p . i = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i)))
proof
let i be Element of NAT ; ::_thesis: ( i in Seg (len D) implies p . i = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i))) )
A4: vol (divset (D,i)) = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i))) ;
assume i in Seg (len D) ; ::_thesis: p . i = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i)))
hence p . i = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i))) by A1, A2, A4; ::_thesis: verum
end;
(len D) - 1 in NAT
proof
ex j being Nat st len D = 1 + j by NAT_1:10, NAT_1:14;
hence (len D) - 1 in NAT by ORDINAL1:def_12; ::_thesis: verum
end;
then reconsider k = (len D) - 1 as Element of NAT ;
deffunc H2( Nat) -> Element of REAL = lower_bound (divset (D,($1 + 1)));
deffunc H3( Nat) -> Element of REAL = upper_bound (divset (D,$1));
consider s1 being FinSequence of REAL such that
A5: ( len s1 = k & ( for i being Nat st i in dom s1 holds
s1 . i = H3(i) ) ) from FINSEQ_2:sch_1();
consider s2 being FinSequence of REAL such that
A6: ( len s2 = k & ( for i being Nat st i in dom s2 holds
s2 . i = H2(i) ) ) from FINSEQ_2:sch_1();
A7: dom s2 = Seg k by A6, FINSEQ_1:def_3;
( len (s1 ^ <*(upper_bound A)*>) = len (<*(lower_bound A)*> ^ s2) & len (s1 ^ <*(upper_bound A)*>) = len p & ( for i being Element of NAT st i in dom (s1 ^ <*(upper_bound A)*>) holds
p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) ) )
proof
dom <*(upper_bound A)*> = Seg 1 by FINSEQ_1:def_8;
then len <*(upper_bound A)*> = 1 by FINSEQ_1:def_3;
then A8: len (s1 ^ <*(upper_bound A)*>) = k + 1 by A5, FINSEQ_1:22;
dom <*(lower_bound A)*> = Seg 1 by FINSEQ_1:def_8;
then len <*(lower_bound A)*> = 1 by FINSEQ_1:def_3;
hence A9: len (s1 ^ <*(upper_bound A)*>) = len (<*(lower_bound A)*> ^ s2) by A6, A8, FINSEQ_1:22; ::_thesis: ( len (s1 ^ <*(upper_bound A)*>) = len p & ( for i being Element of NAT st i in dom (s1 ^ <*(upper_bound A)*>) holds
p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) ) )
thus len (s1 ^ <*(upper_bound A)*>) = len p by A1, A8; ::_thesis: for i being Element of NAT st i in dom (s1 ^ <*(upper_bound A)*>) holds
p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i)
let i be Element of NAT ; ::_thesis: ( i in dom (s1 ^ <*(upper_bound A)*>) implies p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) )
assume A10: i in dom (s1 ^ <*(upper_bound A)*>) ; ::_thesis: p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i)
then A11: (s1 ^ <*(upper_bound A)*>) /. i = (s1 ^ <*(upper_bound A)*>) . i by PARTFUN1:def_6;
i in Seg (len (s1 ^ <*(upper_bound A)*>)) by A10, FINSEQ_1:def_3;
then i in dom (<*(lower_bound A)*> ^ s2) by A9, FINSEQ_1:def_3;
then A12: (<*(lower_bound A)*> ^ s2) /. i = (<*(lower_bound A)*> ^ s2) . i by PARTFUN1:def_6;
A13: ( len D = 1 or not len D is trivial ) by NAT_2:def_1;
now__::_thesis:_p_._i_=_((s1_^_<*(upper_bound_A)*>)_/._i)_-_((<*(lower_bound_A)*>_^_s2)_/._i)
percases ( len D = 1 or len D >= 2 ) by A13, NAT_2:29;
supposeA14: len D = 1 ; ::_thesis: p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i)
then A15: i in Seg 1 by A8, A10, FINSEQ_1:def_3;
then A16: i = 1 by FINSEQ_1:2, TARSKI:def_1;
s1 = {} by A5, A14;
then s1 ^ <*(upper_bound A)*> = <*(upper_bound A)*> by FINSEQ_1:34;
then A17: (s1 ^ <*(upper_bound A)*>) /. i = upper_bound A by A11, A16, FINSEQ_1:def_8;
A18: i in dom D by A14, A15, FINSEQ_1:def_3;
s2 = {} by A6, A14;
then <*(lower_bound A)*> ^ s2 = <*(lower_bound A)*> by FINSEQ_1:34;
then A19: (<*(lower_bound A)*> ^ s2) /. i = lower_bound A by A12, A16, FINSEQ_1:def_8;
D . i = upper_bound A by A14, A16, Def2;
then A20: upper_bound (divset (D,i)) = upper_bound A by A16, A18, Def4;
p . i = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i))) by A3, A14, A15;
hence p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) by A16, A18, A17, A19, A20, Def4; ::_thesis: verum
end;
supposeA21: len D >= 2 ; ::_thesis: p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i)
1 = 2 - 1 ;
then A22: k >= 1 by A21, XREAL_1:9;
now__::_thesis:_p_._i_=_((s1_^_<*(upper_bound_A)*>)_/._i)_-_((<*(lower_bound_A)*>_^_s2)_/._i)
percases ( i = 1 or i = len D or ( i <> 1 & i <> len D ) ) ;
supposeA23: i = 1 ; ::_thesis: p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i)
then A24: i in Seg 1 by FINSEQ_1:2, TARSKI:def_1;
then i in dom <*(lower_bound A)*> by FINSEQ_1:def_8;
then (<*(lower_bound A)*> ^ s2) . i = <*(lower_bound A)*> . i by FINSEQ_1:def_7;
then A25: (<*(lower_bound A)*> ^ s2) . i = lower_bound A by A23, FINSEQ_1:def_8;
Seg 1 c= Seg k by A22, FINSEQ_1:5;
then i in Seg k by A24;
then A26: i in dom s1 by A5, FINSEQ_1:def_3;
then (s1 ^ <*(upper_bound A)*>) . i = s1 . i by FINSEQ_1:def_7;
then A27: (s1 ^ <*(upper_bound A)*>) . i = upper_bound (divset (D,i)) by A5, A26;
A28: i in Seg 2 by A23, FINSEQ_1:2, TARSKI:def_2;
A29: Seg 2 c= Seg (len D) by A21, FINSEQ_1:5;
then i in Seg (len D) by A28;
then A30: i in dom D by FINSEQ_1:def_3;
p . i = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i))) by A3, A29, A28;
hence p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) by A11, A12, A23, A30, A27, A25, Def4; ::_thesis: verum
end;
supposeA31: i = len D ; ::_thesis: p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i)
then i - (len s1) in Seg 1 by A5, FINSEQ_1:2, TARSKI:def_1;
then A32: i - (len s1) in dom <*(upper_bound A)*> by FINSEQ_1:def_8;
i = (i - (len s1)) + (len s1) ;
then (s1 ^ <*(upper_bound A)*>) . i = <*(upper_bound A)*> . (i - (len s1)) by A32, FINSEQ_1:def_7;
then A33: (s1 ^ <*(upper_bound A)*>) /. i = upper_bound A by A5, A11, A31, FINSEQ_1:def_8;
A34: i <> 1 by A21, A31;
i in Seg (len D) by A31, FINSEQ_1:3;
then A35: i in dom D by FINSEQ_1:def_3;
p . i = (upper_bound (divset (D,i))) - (lower_bound (divset (D,i))) by A3, A31, FINSEQ_1:3;
then p . i = (upper_bound (divset (D,i))) - (D . (i - 1)) by A35, A34, Def4;
then A36: p . i = (D . i) - (D . (i - 1)) by A35, A34, Def4;
A37: i - (len <*(lower_bound A)*>) = i - 1 by FINSEQ_1:40;
i - 1 <> 0 by A21, A31;
then i - 1 in Seg k by A31, FINSEQ_1:3;
then A38: i - (len <*(lower_bound A)*>) in dom s2 by A6, A37, FINSEQ_1:def_3;
A39: (len <*(lower_bound A)*>) + (i - (len <*(lower_bound A)*>)) = i ;
then (<*(lower_bound A)*> ^ s2) . i = s2 . (i - (len <*(lower_bound A)*>)) by A38, FINSEQ_1:def_7;
then (<*(lower_bound A)*> ^ s2) . i = lower_bound (divset (D,i)) by A6, A37, A39, A38;
then (<*(lower_bound A)*> ^ s2) . i = D . (i - 1) by A35, A34, Def4;
hence p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) by A12, A31, A33, A36, Def2; ::_thesis: verum
end;
supposeA40: ( i <> 1 & i <> len D ) ; ::_thesis: p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i)
(len s1) + (len <*(upper_bound A)*>) = k + 1 by A5, FINSEQ_1:39;
then A41: i in Seg (len D) by A10, FINSEQ_1:def_7;
A42: ( i in dom s1 & i in Seg k & i - 1 in Seg k & (i - 1) + 1 = i & i - (len <*(lower_bound A)*>) in dom s2 )
proof
i <> 0 by A41, FINSEQ_1:1;
then not i is trivial by A40, NAT_2:def_1;
then i >= 1 + 1 by NAT_2:29;
then A43: i - 1 >= 1 by XREAL_1:19;
A44: 1 <= i by A41, FINSEQ_1:1;
i <= len D by A41, FINSEQ_1:1;
then A45: i < k + 1 by A40, XXREAL_0:1;
then A46: i <= k by NAT_1:13;
then i in Seg (len s1) by A5, A44, FINSEQ_1:1;
hence i in dom s1 by FINSEQ_1:def_3; ::_thesis: ( i in Seg k & i - 1 in Seg k & (i - 1) + 1 = i & i - (len <*(lower_bound A)*>) in dom s2 )
thus i in Seg k by A44, A46, FINSEQ_1:1; ::_thesis: ( i - 1 in Seg k & (i - 1) + 1 = i & i - (len <*(lower_bound A)*>) in dom s2 )
i <= k by A45, NAT_1:13;
then i - 1 <= k - 1 by XREAL_1:9;
then A47: (i - 1) + 0 <= (k - 1) + 1 by XREAL_1:7;
ex j being Nat st i = 1 + j by A44, NAT_1:10;
hence i - 1 in Seg k by A43, A47, FINSEQ_1:1; ::_thesis: ( (i - 1) + 1 = i & i - (len <*(lower_bound A)*>) in dom s2 )
then A48: i - (len <*(lower_bound A)*>) in Seg (len s2) by A6, FINSEQ_1:39;
thus (i - 1) + 1 = i ; ::_thesis: i - (len <*(lower_bound A)*>) in dom s2
thus i - (len <*(lower_bound A)*>) in dom s2 by A48, FINSEQ_1:def_3; ::_thesis: verum
end;
then A49: i - (len <*(lower_bound A)*>) in Seg (len s2) by FINSEQ_1:def_3;
then i - (len <*(lower_bound A)*>) <= len s2 by FINSEQ_1:1;
then A50: i <= (len <*(lower_bound A)*>) + (len s2) by XREAL_1:20;
1 <= i - (len <*(lower_bound A)*>) by A49, FINSEQ_1:1;
then (len <*(lower_bound A)*>) + 1 <= i by XREAL_1:19;
then (<*(lower_bound A)*> ^ s2) . i = s2 . (i - (len <*(lower_bound A)*>)) by A50, FINSEQ_1:23;
then (<*(lower_bound A)*> ^ s2) . i = s2 . (i - 1) by FINSEQ_1:39;
then A51: (<*(lower_bound A)*> ^ s2) . i = lower_bound (divset (D,i)) by A6, A7, A42;
(s1 ^ <*(upper_bound A)*>) . i = s1 . i by A42, FINSEQ_1:def_7;
then (s1 ^ <*(upper_bound A)*>) . i = upper_bound (divset (D,i)) by A5, A42;
hence p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) by A3, A11, A12, A41, A51; ::_thesis: verum
end;
end;
end;
hence p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) ; ::_thesis: verum
end;
end;
end;
hence p . i = ((s1 ^ <*(upper_bound A)*>) /. i) - ((<*(lower_bound A)*> ^ s2) /. i) ; ::_thesis: verum
end;
then Sum p = (Sum (s1 ^ <*(upper_bound A)*>)) - (Sum (<*(lower_bound A)*> ^ s2)) by Th22;
then Sum p = ((Sum s1) + (upper_bound A)) - (Sum (<*(lower_bound A)*> ^ s2)) by RVSUM_1:74;
then A52: Sum p = ((Sum s1) + (upper_bound A)) - ((lower_bound A) + (Sum s2)) by RVSUM_1:76;
A53: dom s1 = Seg k by A5, FINSEQ_1:def_3;
A54: for i being Element of NAT st i in Seg k holds
upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1)))
proof
let i be Element of NAT ; ::_thesis: ( i in Seg k implies upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1))) )
A55: 1 + 0 <= i + 1 by XREAL_1:7;
assume A56: i in Seg k ; ::_thesis: upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1)))
then i <= k by FINSEQ_1:1;
then i + 1 <= k + 1 by XREAL_1:7;
then i + 1 in Seg (len D) by A55, FINSEQ_1:1;
then A57: i + 1 in dom D by FINSEQ_1:def_3;
k + 1 = len D ;
then k <= len D by NAT_1:11;
then Seg k c= Seg (len D) by FINSEQ_1:5;
then i in Seg (len D) by A56;
then A58: i in dom D by FINSEQ_1:def_3;
A59: (i + 1) - 1 = i ;
now__::_thesis:_upper_bound_(divset_(D,i))_=_lower_bound_(divset_(D,(i_+_1)))
percases ( i = 1 or i <> 1 ) ;
supposeA60: i = 1 ; ::_thesis: upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1)))
then upper_bound (divset (D,i)) = D . i by A58, Def4;
hence upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1))) by A57, A59, A60, Def4; ::_thesis: verum
end;
supposeA61: i <> 1 ; ::_thesis: upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1)))
i >= 1 by A56, FINSEQ_1:1;
then i + 1 >= 1 + 1 by XREAL_1:6;
then A62: i + 1 <> 1 ;
upper_bound (divset (D,i)) = D . i by A58, A61, Def4;
hence upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1))) by A57, A59, A62, Def4; ::_thesis: verum
end;
end;
end;
hence upper_bound (divset (D,i)) = lower_bound (divset (D,(i + 1))) ; ::_thesis: verum
end;
for i being Nat st i in dom s1 holds
s1 . i = s2 . i
proof
let i be Nat; ::_thesis: ( i in dom s1 implies s1 . i = s2 . i )
assume A63: i in dom s1 ; ::_thesis: s1 . i = s2 . i
then s1 . i = upper_bound (divset (D,i)) by A5;
then s1 . i = lower_bound (divset (D,(i + 1))) by A54, A53, A63;
hence s1 . i = s2 . i by A6, A7, A53, A63; ::_thesis: verum
end;
then A64: s1 = s2 by A5, A6, FINSEQ_2:9;
A65: len (lower_volume ((chi (A,A)),D)) = len D by Def7;
then A66: dom (lower_volume ((chi (A,A)),D)) = Seg (len D) by FINSEQ_1:def_3;
for i being Nat st i in dom (lower_volume ((chi (A,A)),D)) holds
(lower_volume ((chi (A,A)),D)) . i = p . i
proof
let i be Nat; ::_thesis: ( i in dom (lower_volume ((chi (A,A)),D)) implies (lower_volume ((chi (A,A)),D)) . i = p . i )
assume A67: i in dom (lower_volume ((chi (A,A)),D)) ; ::_thesis: (lower_volume ((chi (A,A)),D)) . i = p . i
then i in dom D by A65, FINSEQ_3:29;
then (lower_volume ((chi (A,A)),D)) . i = vol (divset (D,i)) by Th19;
hence (lower_volume ((chi (A,A)),D)) . i = p . i by A1, A2, A66, A67; ::_thesis: verum
end;
hence Sum (lower_volume ((chi (A,A)),D)) = vol A by A1, A64, A52, A65, FINSEQ_2:9; ::_thesis: verum
end;
theorem Th24: :: INTEGRA1:24
for A being non empty closed_interval Subset of REAL
for D being Division of A holds Sum (upper_volume ((chi (A,A)),D)) = vol A
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A holds Sum (upper_volume ((chi (A,A)),D)) = vol A
let D be Division of A; ::_thesis: Sum (upper_volume ((chi (A,A)),D)) = vol A
A1: for i being Nat st 1 <= i & i <= len (lower_volume ((chi (A,A)),D)) holds
(lower_volume ((chi (A,A)),D)) . i = (upper_volume ((chi (A,A)),D)) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len (lower_volume ((chi (A,A)),D)) implies (lower_volume ((chi (A,A)),D)) . i = (upper_volume ((chi (A,A)),D)) . i )
assume that
A2: 1 <= i and
A3: i <= len (lower_volume ((chi (A,A)),D)) ; ::_thesis: (lower_volume ((chi (A,A)),D)) . i = (upper_volume ((chi (A,A)),D)) . i
i <= len D by A3, Def7;
then A4: i in dom D by A2, FINSEQ_3:25;
then (lower_volume ((chi (A,A)),D)) . i = vol (divset (D,i)) by Th19
.= (upper_volume ((chi (A,A)),D)) . i by A4, Th20 ;
hence (lower_volume ((chi (A,A)),D)) . i = (upper_volume ((chi (A,A)),D)) . i ; ::_thesis: verum
end;
len (lower_volume ((chi (A,A)),D)) = len D by Def7
.= len (upper_volume ((chi (A,A)),D)) by Def6 ;
then lower_volume ((chi (A,A)),D) = upper_volume ((chi (A,A)),D) by A1, FINSEQ_1:14;
hence Sum (upper_volume ((chi (A,A)),D)) = vol A by Th23; ::_thesis: verum
end;
begin
registration
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of A,REAL;
let D be Division of A;
cluster upper_volume (f,D) -> non empty ;
coherence
not upper_volume (f,D) is empty
proof
len (upper_volume (f,D)) = len D by Def6;
hence not upper_volume (f,D) is empty ; ::_thesis: verum
end;
cluster lower_volume (f,D) -> non empty ;
coherence
not lower_volume (f,D) is empty
proof
len (lower_volume (f,D)) = len D by Def7;
hence not lower_volume (f,D) is empty ; ::_thesis: verum
end;
end;
theorem Th25: :: INTEGRA1:25
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_below holds
(lower_bound (rng f)) * (vol A) <= lower_sum (f,D)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL st f | A is bounded_below holds
(lower_bound (rng f)) * (vol A) <= lower_sum (f,D)
let D be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded_below holds
(lower_bound (rng f)) * (vol A) <= lower_sum (f,D)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded_below implies (lower_bound (rng f)) * (vol A) <= lower_sum (f,D) )
assume A1: f | A is bounded_below ; ::_thesis: (lower_bound (rng f)) * (vol A) <= lower_sum (f,D)
A2: for i being Element of NAT st i in dom D holds
(lower_bound (rng f)) * (vol (divset (D,i))) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
proof
let i be Element of NAT ; ::_thesis: ( i in dom D implies (lower_bound (rng f)) * (vol (divset (D,i))) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
A3: rng (f | (divset (D,i))) c= rng f by RELAT_1:70;
A4: 0 <= vol (divset (D,i)) by SEQ_4:11, XREAL_1:48;
A5: dom f = A by FUNCT_2:def_1;
assume i in dom D ; ::_thesis: (lower_bound (rng f)) * (vol (divset (D,i))) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
then dom (f | (divset (D,i))) = divset (D,i) by A5, Th8, RELAT_1:62;
then A6: rng (f | (divset (D,i))) is non empty Subset of REAL by RELAT_1:42;
rng f is bounded_below by A1, Th11;
hence (lower_bound (rng f)) * (vol (divset (D,i))) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A3, A6, A4, SEQ_4:47, XREAL_1:64; ::_thesis: verum
end;
A7: for i being Element of NAT st i in dom D holds
(lower_bound (rng f)) * ((lower_volume ((chi (A,A)),D)) . i) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
proof
let i be Element of NAT ; ::_thesis: ( i in dom D implies (lower_bound (rng f)) * ((lower_volume ((chi (A,A)),D)) . i) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
assume A8: i in dom D ; ::_thesis: (lower_bound (rng f)) * ((lower_volume ((chi (A,A)),D)) . i) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
then (lower_bound (rng f)) * (vol (divset (D,i))) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A2;
hence (lower_bound (rng f)) * ((lower_volume ((chi (A,A)),D)) . i) <= (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A8, Th19; ::_thesis: verum
end;
Sum ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) <= Sum (lower_volume (f,D))
proof
len (lower_volume ((chi (A,A)),D)) = len ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) by FINSEQ_2:33;
then A9: len ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) = len D by Def7;
deffunc H1( Nat) -> Element of REAL = (lower_bound (rng (f | (divset (D,$1))))) * (vol (divset (D,$1)));
deffunc H2( set ) -> Element of REAL = (lower_bound (rng f)) * ((lower_volume ((chi (A,A)),D)) . $1);
consider p being FinSequence of REAL such that
A10: ( len p = len D & ( for i being Nat st i in dom p holds
p . i = H2(i) ) ) from FINSEQ_2:sch_1();
A11: dom p = Seg (len D) by A10, FINSEQ_1:def_3;
for i being Nat st 1 <= i & i <= len p holds
p . i = ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len p implies p . i = ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) . i )
assume that
A12: 1 <= i and
A13: i <= len p ; ::_thesis: p . i = ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) . i
i in Seg (len D) by A10, A12, A13, FINSEQ_1:1;
then p . i = (lower_bound (rng f)) * ((lower_volume ((chi (A,A)),D)) . i) by A10, A11;
hence p . i = ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) . i by RVSUM_1:44; ::_thesis: verum
end;
then A14: p = (lower_bound (rng f)) * (lower_volume ((chi (A,A)),D)) by A10, A9, FINSEQ_1:14;
reconsider p = p as Element of (len D) -tuples_on REAL by A10, FINSEQ_2:92;
consider q being FinSequence of REAL such that
A15: ( len q = len D & ( for i being Nat st i in dom q holds
q . i = H1(i) ) ) from FINSEQ_2:sch_1();
A16: dom q = dom D by A15, FINSEQ_3:29;
then A17: q = lower_volume (f,D) by A15, Def7;
reconsider q = q as Element of (len D) -tuples_on REAL by A15, FINSEQ_2:92;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_(len_D)_holds_
p_._i_<=_q_._i
let i be Nat; ::_thesis: ( i in Seg (len D) implies p . i <= q . i )
assume A18: i in Seg (len D) ; ::_thesis: p . i <= q . i
then A19: p . i = (lower_bound (rng f)) * ((lower_volume ((chi (A,A)),D)) . i) by A10, A11;
A20: i in dom D by A18, FINSEQ_1:def_3;
then q . i = (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A15, A16;
hence p . i <= q . i by A7, A19, A20; ::_thesis: verum
end;
hence Sum ((lower_bound (rng f)) * (lower_volume ((chi (A,A)),D))) <= Sum (lower_volume (f,D)) by A17, A14, RVSUM_1:82; ::_thesis: verum
end;
then (lower_bound (rng f)) * (Sum (lower_volume ((chi (A,A)),D))) <= Sum (lower_volume (f,D)) by RVSUM_1:87;
hence (lower_bound (rng f)) * (vol A) <= lower_sum (f,D) by Th23; ::_thesis: verum
end;
theorem :: INTEGRA1:26
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
let D be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded_above & i in dom D holds
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
let f be Function of A,REAL; ::_thesis: ( f | A is bounded_above & i in dom D implies (upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
A1: dom f = A by FUNCT_2:def_1;
assume f | A is bounded_above ; ::_thesis: ( not i in dom D or (upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
then A2: rng f is bounded_above by Th13;
assume i in dom D ; ::_thesis: (upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
then dom (f | (divset (D,i))) = divset (D,i) by A1, Th8, RELAT_1:62;
then A3: rng (f | (divset (D,i))) is non empty Subset of REAL by RELAT_1:42;
A4: 0 <= vol (divset (D,i)) by SEQ_4:11, XREAL_1:48;
rng (f | (divset (D,i))) c= rng f by RELAT_1:70;
hence (upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A3, A2, A4, SEQ_4:48, XREAL_1:64; ::_thesis: verum
end;
theorem Th27: :: INTEGRA1:27
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded_above holds
upper_sum (f,D) <= (upper_bound (rng f)) * (vol A)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL st f | A is bounded_above holds
upper_sum (f,D) <= (upper_bound (rng f)) * (vol A)
let D be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded_above holds
upper_sum (f,D) <= (upper_bound (rng f)) * (vol A)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded_above implies upper_sum (f,D) <= (upper_bound (rng f)) * (vol A) )
assume A1: f | A is bounded_above ; ::_thesis: upper_sum (f,D) <= (upper_bound (rng f)) * (vol A)
A2: for i being Element of NAT st i in Seg (len D) holds
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
proof
let i be Element of NAT ; ::_thesis: ( i in Seg (len D) implies (upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
A3: rng (f | (divset (D,i))) c= rng f by RELAT_1:70;
A4: 0 <= vol (divset (D,i)) by SEQ_4:11, XREAL_1:48;
assume i in Seg (len D) ; ::_thesis: (upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
then A5: i in dom D by FINSEQ_1:def_3;
dom f = A by FUNCT_2:def_1;
then dom (f | (divset (D,i))) = divset (D,i) by A5, Th8, RELAT_1:62;
then A6: rng (f | (divset (D,i))) is non empty Subset of REAL by RELAT_1:42;
rng f is bounded_above by A1, Th13;
hence (upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A3, A6, A4, SEQ_4:48, XREAL_1:64; ::_thesis: verum
end;
A7: for i being Element of NAT st i in Seg (len D) holds
(upper_bound (rng f)) * ((upper_volume ((chi (A,A)),D)) . i) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
proof
let i be Element of NAT ; ::_thesis: ( i in Seg (len D) implies (upper_bound (rng f)) * ((upper_volume ((chi (A,A)),D)) . i) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) )
assume A8: i in Seg (len D) ; ::_thesis: (upper_bound (rng f)) * ((upper_volume ((chi (A,A)),D)) . i) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))
then A9: i in dom D by FINSEQ_1:def_3;
(upper_bound (rng f)) * (vol (divset (D,i))) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A2, A8;
hence (upper_bound (rng f)) * ((upper_volume ((chi (A,A)),D)) . i) >= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A9, Th20; ::_thesis: verum
end;
Sum ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) >= Sum (upper_volume (f,D))
proof
len (upper_volume ((chi (A,A)),D)) = len ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) by FINSEQ_2:33;
then A10: len ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) = len D by Def6;
deffunc H1( Nat) -> Element of REAL = (upper_bound (rng (f | (divset (D,$1))))) * (vol (divset (D,$1)));
deffunc H2( set ) -> Element of REAL = (upper_bound (rng f)) * ((upper_volume ((chi (A,A)),D)) . $1);
consider p being FinSequence of REAL such that
A11: ( len p = len D & ( for i being Nat st i in dom p holds
p . i = H2(i) ) ) from FINSEQ_2:sch_1();
A12: dom p = Seg (len D) by A11, FINSEQ_1:def_3;
for i being Nat st 1 <= i & i <= len p holds
p . i = ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len p implies p . i = ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) . i )
assume that
A13: 1 <= i and
A14: i <= len p ; ::_thesis: p . i = ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) . i
i in Seg (len D) by A11, A13, A14, FINSEQ_1:1;
then p . i = (upper_bound (rng f)) * ((upper_volume ((chi (A,A)),D)) . i) by A11, A12;
hence p . i = ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) . i by RVSUM_1:44; ::_thesis: verum
end;
then A15: p = (upper_bound (rng f)) * (upper_volume ((chi (A,A)),D)) by A11, A10, FINSEQ_1:14;
reconsider p = p as Element of (len D) -tuples_on REAL by A11, FINSEQ_2:92;
consider q being FinSequence of REAL such that
A16: ( len q = len D & ( for i being Nat st i in dom q holds
q . i = H1(i) ) ) from FINSEQ_2:sch_1();
A17: dom q = dom D by A16, FINSEQ_3:29;
then A18: q = upper_volume (f,D) by A16, Def6;
reconsider q = q as Element of (len D) -tuples_on REAL by A16, FINSEQ_2:92;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_(len_D)_holds_
q_._i_<=_p_._i
let i be Nat; ::_thesis: ( i in Seg (len D) implies q . i <= p . i )
assume A19: i in Seg (len D) ; ::_thesis: q . i <= p . i
then i in dom D by FINSEQ_1:def_3;
then A20: q . i = (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A16, A17;
p . i = (upper_bound (rng f)) * ((upper_volume ((chi (A,A)),D)) . i) by A11, A12, A19;
hence q . i <= p . i by A7, A19, A20; ::_thesis: verum
end;
hence Sum ((upper_bound (rng f)) * (upper_volume ((chi (A,A)),D))) >= Sum (upper_volume (f,D)) by A18, A15, RVSUM_1:82; ::_thesis: verum
end;
then (upper_bound (rng f)) * (Sum (upper_volume ((chi (A,A)),D))) >= Sum (upper_volume (f,D)) by RVSUM_1:87;
hence upper_sum (f,D) <= (upper_bound (rng f)) * (vol A) by Th24; ::_thesis: verum
end;
theorem Th28: :: INTEGRA1:28
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D) <= upper_sum (f,D)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D) <= upper_sum (f,D)
let D be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D) <= upper_sum (f,D)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded implies lower_sum (f,D) <= upper_sum (f,D) )
deffunc H1( Nat) -> Element of REAL = (lower_bound (rng (f | (divset (D,$1))))) * (vol (divset (D,$1)));
consider p being FinSequence of REAL such that
A1: ( len p = len D & ( for i being Nat st i in dom p holds
p . i = H1(i) ) ) from FINSEQ_2:sch_1();
assume A2: f | A is bounded ; ::_thesis: lower_sum (f,D) <= upper_sum (f,D)
then A3: rng f is bounded_above by Th13;
A4: dom p = dom D by A1, FINSEQ_3:29;
reconsider p = p as Element of (len D) -tuples_on REAL by A1, FINSEQ_2:92;
deffunc H2( Nat) -> Element of REAL = (upper_bound (rng (f | (divset (D,$1))))) * (vol (divset (D,$1)));
consider q being FinSequence of REAL such that
A5: ( len q = len D & ( for i being Nat st i in dom q holds
q . i = H2(i) ) ) from FINSEQ_2:sch_1();
A6: dom q = dom D by A5, FINSEQ_3:29;
then A7: q = upper_volume (f,D) by A5, Def6;
reconsider q = q as Element of (len D) -tuples_on REAL by A5, FINSEQ_2:92;
A8: rng f is bounded_below by A2, Th11;
for i being Nat st i in Seg (len D) holds
p . i <= q . i
proof
let i be Nat; ::_thesis: ( i in Seg (len D) implies p . i <= q . i )
A9: dom f = A by FUNCT_2:def_1;
assume A10: i in Seg (len D) ; ::_thesis: p . i <= q . i
then A11: i in dom D by FINSEQ_1:def_3;
i in dom D by A10, FINSEQ_1:def_3;
then dom (f | (divset (D,i))) = divset (D,i) by A9, Th8, RELAT_1:62;
then A12: rng (f | (divset (D,i))) is non empty Subset of REAL by RELAT_1:42;
A13: 0 <= vol (divset (D,i)) by SEQ_4:11, XREAL_1:48;
A14: rng (f | (divset (D,i))) is bounded_above by A3, RELAT_1:70, XXREAL_2:43;
rng (f | (divset (D,i))) is bounded_below by A8, RELAT_1:70, XXREAL_2:44;
then (lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) <= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A14, A12, A13, SEQ_4:11, XREAL_1:64;
then p . i <= (upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i))) by A1, A4, A11;
hence p . i <= q . i by A5, A6, A11; ::_thesis: verum
end;
then Sum p <= Sum q by RVSUM_1:82;
hence lower_sum (f,D) <= upper_sum (f,D) by A1, A4, A7, Def7; ::_thesis: verum
end;
definition
let D1, D2 be FinSequence;
predD1 <= D2 means :Def18: :: INTEGRA1:def 18
( len D1 <= len D2 & rng D1 c= rng D2 );
reflexivity
for D1 being FinSequence holds
( len D1 <= len D1 & rng D1 c= rng D1 ) ;
end;
:: deftheorem Def18 defines <= INTEGRA1:def_18_:_
for D1, D2 being FinSequence holds
( D1 <= D2 iff ( len D1 <= len D2 & rng D1 c= rng D2 ) );
notation
let D1, D2 be FinSequence;
synonym D2 >= D1 for D1 <= D2;
end;
theorem :: INTEGRA1:29
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st len D1 = 1 holds
D1 <= D2
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A st len D1 = 1 holds
D1 <= D2
let D1, D2 be Division of A; ::_thesis: ( len D1 = 1 implies D1 <= D2 )
A1: D2 . (len D2) = upper_bound A by Def2;
assume A2: len D1 = 1 ; ::_thesis: D1 <= D2
then D1 . 1 = upper_bound A by Def2;
then D1 = <*(upper_bound A)*> by A2, FINSEQ_1:40;
then A3: rng D1 = {(upper_bound A)} by FINSEQ_1:38;
A4: len D2 in Seg (len D2) by FINSEQ_1:3;
hence len D1 <= len D2 by A2, FINSEQ_1:1; :: according to INTEGRA1:def_18 ::_thesis: rng D1 c= rng D2
len D2 in dom D2 by A4, FINSEQ_1:def_3;
then upper_bound A in rng D2 by A1, FUNCT_1:def_3;
then rng D1 is Subset of (rng D2) by A3, SUBSET_1:41;
hence rng D1 c= rng D2 ; ::_thesis: verum
end;
theorem Th30: :: INTEGRA1:30
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded_above & len D1 = 1 holds
upper_sum (f,D1) >= upper_sum (f,D2)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded_above & len D1 = 1 holds
upper_sum (f,D1) >= upper_sum (f,D2)
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded_above & len D1 = 1 holds
upper_sum (f,D1) >= upper_sum (f,D2)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded_above & len D1 = 1 implies upper_sum (f,D1) >= upper_sum (f,D2) )
assume that
A1: f | A is bounded_above and
A2: len D1 = 1 ; ::_thesis: upper_sum (f,D1) >= upper_sum (f,D2)
1 in Seg (len D1) by A2, FINSEQ_1:3;
then A3: 1 in dom D1 by FINSEQ_1:def_3;
then A4: lower_bound (divset (D1,1)) = lower_bound A by Def4;
A5: divset (D1,1) = [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by Th4;
upper_bound (divset (D1,1)) = D1 . 1 by A3, Def4
.= upper_bound A by A2, Def2 ;
then A6: divset (D1,1) = A by A4, A5, Th4;
A7: (upper_volume (f,D1)) . 1 = (upper_bound (rng (f | (divset (D1,1))))) * (vol (divset (D1,1))) by A3, Def6;
len (upper_volume (f,D1)) = 1 by A2, Def6;
then upper_sum (f,D1) = Sum <*((upper_bound (rng (f | (divset (D1,1))))) * (vol (divset (D1,1))))*> by A7, FINSEQ_1:40
.= (upper_bound (rng (f | A))) * (vol A) by A6, FINSOP_1:11
.= (upper_bound (rng f)) * (vol A) ;
hence upper_sum (f,D1) >= upper_sum (f,D2) by A1, Th27; ::_thesis: verum
end;
theorem Th31: :: INTEGRA1:31
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded_below & len D1 = 1 holds
lower_sum (f,D1) <= lower_sum (f,D2)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded_below & len D1 = 1 holds
lower_sum (f,D1) <= lower_sum (f,D2)
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded_below & len D1 = 1 holds
lower_sum (f,D1) <= lower_sum (f,D2)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded_below & len D1 = 1 implies lower_sum (f,D1) <= lower_sum (f,D2) )
assume that
A1: f | A is bounded_below and
A2: len D1 = 1 ; ::_thesis: lower_sum (f,D1) <= lower_sum (f,D2)
1 in Seg (len D1) by A2, FINSEQ_1:3;
then A3: 1 in dom D1 by FINSEQ_1:def_3;
then A4: lower_bound (divset (D1,1)) = lower_bound A by Def4;
upper_bound (divset (D1,1)) = D1 . 1 by A3, Def4
.= upper_bound A by A2, Def2 ;
then divset (D1,1) = [.(lower_bound A),(upper_bound A).] by A4, Th4;
then A5: divset (D1,1) = A by Th4;
A6: (lower_volume (f,D1)) . 1 = (lower_bound (rng (f | (divset (D1,1))))) * (vol (divset (D1,1))) by A3, Def7;
len (lower_volume (f,D1)) = 1 by A2, Def7;
then lower_sum (f,D1) = Sum <*((lower_bound (rng (f | (divset (D1,1))))) * (vol (divset (D1,1))))*> by A6, FINSEQ_1:40
.= (lower_bound (rng (f | A))) * (vol A) by A5, FINSOP_1:11
.= (lower_bound (rng f)) * (vol A) ;
hence lower_sum (f,D1) <= lower_sum (f,D2) by A1, Th25; ::_thesis: verum
end;
theorem :: INTEGRA1:32
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
ex A1, A2 being non empty closed_interval Subset of REAL st
( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 )
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D holds
ex A1, A2 being non empty closed_interval Subset of REAL st
( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D holds
ex A1, A2 being non empty closed_interval Subset of REAL st
( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 )
let D be Division of A; ::_thesis: ( i in dom D implies ex A1, A2 being non empty closed_interval Subset of REAL st
( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 ) )
assume i in dom D ; ::_thesis: ex A1, A2 being non empty closed_interval Subset of REAL st
( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 )
then A1: D . i in rng D by FUNCT_1:def_3;
rng D c= A by Def2;
then D . i in A by A1;
then D . i in [.(lower_bound A),(upper_bound A).] by Th4;
then D . i in { a where a is Real : ( lower_bound A <= a & a <= upper_bound A ) } by RCOMP_1:def_1;
then A2: ex a being Real st
( a = D . i & lower_bound A <= a & a <= upper_bound A ) ;
then reconsider A1 = [.(lower_bound A),(D . i).] as non empty closed_interval Subset of REAL by MEASURE5:14;
reconsider A2 = [.(D . i),(upper_bound A).] as non empty closed_interval Subset of REAL by A2, MEASURE5:14;
take A1 ; ::_thesis: ex A2 being non empty closed_interval Subset of REAL st
( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 )
take A2 ; ::_thesis: ( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 )
A1 \/ A2 = [.(lower_bound A),(upper_bound A).] by A2, XXREAL_1:174
.= A by Th4 ;
hence ( A1 = [.(lower_bound A),(D . i).] & A2 = [.(D . i),(upper_bound A).] & A = A1 \/ A2 ) ; ::_thesis: verum
end;
theorem Th33: :: INTEGRA1:33
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st i in dom D1 & D1 <= D2 holds
ex j being Element of NAT st
( j in dom D2 & D1 . i = D2 . j )
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st i in dom D1 & D1 <= D2 holds
ex j being Element of NAT st
( j in dom D2 & D1 . i = D2 . j )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A st i in dom D1 & D1 <= D2 holds
ex j being Element of NAT st
( j in dom D2 & D1 . i = D2 . j )
let D1, D2 be Division of A; ::_thesis: ( i in dom D1 & D1 <= D2 implies ex j being Element of NAT st
( j in dom D2 & D1 . i = D2 . j ) )
assume i in dom D1 ; ::_thesis: ( not D1 <= D2 or ex j being Element of NAT st
( j in dom D2 & D1 . i = D2 . j ) )
then A1: D1 . i in rng D1 by FUNCT_1:def_3;
assume D1 <= D2 ; ::_thesis: ex j being Element of NAT st
( j in dom D2 & D1 . i = D2 . j )
then rng D1 c= rng D2 by Def18;
then consider x1 being set such that
A2: x1 in dom D2 and
A3: D1 . i = D2 . x1 by A1, FUNCT_1:def_3;
reconsider x1 = x1 as Element of NAT by A2;
take x1 ; ::_thesis: ( x1 in dom D2 & D1 . i = D2 . x1 )
thus ( x1 in dom D2 & D1 . i = D2 . x1 ) by A2, A3; ::_thesis: verum
end;
definition
let A be non empty closed_interval Subset of REAL;
let D1, D2 be Division of A;
let i be Nat;
assume A1: D1 <= D2 ;
func indx (D2,D1,i) -> Element of NAT means :Def19: :: INTEGRA1:def 19
( it in dom D2 & D1 . i = D2 . it ) if i in dom D1
otherwise it = 0 ;
existence
( ( i in dom D1 implies ex b1 being Element of NAT st
( b1 in dom D2 & D1 . i = D2 . b1 ) ) & ( not i in dom D1 implies ex b1 being Element of NAT st b1 = 0 ) ) by A1, Th33;
uniqueness
for b1, b2 being Element of NAT holds
( ( i in dom D1 & b1 in dom D2 & D1 . i = D2 . b1 & b2 in dom D2 & D1 . i = D2 . b2 implies b1 = b2 ) & ( not i in dom D1 & b1 = 0 & b2 = 0 implies b1 = b2 ) )
proof
let m, n be Element of NAT ; ::_thesis: ( ( i in dom D1 & m in dom D2 & D1 . i = D2 . m & n in dom D2 & D1 . i = D2 . n implies m = n ) & ( not i in dom D1 & m = 0 & n = 0 implies m = n ) )
hereby ::_thesis: ( not i in dom D1 & m = 0 & n = 0 implies m = n )
assume that
i in dom D1 and
A2: m in dom D2 and
A3: D1 . i = D2 . m and
A4: n in dom D2 and
A5: D1 . i = D2 . n ; ::_thesis: not m <> n
assume A6: m <> n ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( m < n or n < m ) by A6, XXREAL_0:1;
suppose m < n ; ::_thesis: contradiction
hence contradiction by A2, A3, A4, A5, SEQM_3:def_1; ::_thesis: verum
end;
suppose n < m ; ::_thesis: contradiction
hence contradiction by A2, A3, A4, A5, SEQM_3:def_1; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
assume that
not i in dom D1 and
A7: m = 0 and
A8: n = 0 ; ::_thesis: m = n
thus m = n by A7, A8; ::_thesis: verum
end;
correctness
consistency
for b1 being Element of NAT holds verum;
;
end;
:: deftheorem Def19 defines indx INTEGRA1:def_19_:_
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for i being Nat st D1 <= D2 holds
for b5 being Element of NAT holds
( ( i in dom D1 implies ( b5 = indx (D2,D1,i) iff ( b5 in dom D2 & D1 . i = D2 . b5 ) ) ) & ( not i in dom D1 implies ( b5 = indx (D2,D1,i) iff b5 = 0 ) ) );
theorem Th34: :: INTEGRA1:34
for p being increasing FinSequence of REAL
for n being Element of NAT st n <= len p holds
p /^ n is increasing FinSequence of REAL
proof
let p be increasing FinSequence of REAL ; ::_thesis: for n being Element of NAT st n <= len p holds
p /^ n is increasing FinSequence of REAL
let n be Element of NAT ; ::_thesis: ( n <= len p implies p /^ n is increasing FinSequence of REAL )
assume A1: n <= len p ; ::_thesis: p /^ n is increasing FinSequence of REAL
for i, j being Element of NAT st i in dom (p /^ n) & j in dom (p /^ n) & i < j holds
(p /^ n) . i < (p /^ n) . j
proof
let i, j be Element of NAT ; ::_thesis: ( i in dom (p /^ n) & j in dom (p /^ n) & i < j implies (p /^ n) . i < (p /^ n) . j )
assume that
A2: i in dom (p /^ n) and
A3: j in dom (p /^ n) and
A4: i < j ; ::_thesis: (p /^ n) . i < (p /^ n) . j
A5: i + n < j + n by A4, XREAL_1:6;
A6: j in Seg (len (p /^ n)) by A3, FINSEQ_1:def_3;
then 1 <= j by FINSEQ_1:1;
then A7: 1 + n <= j + n by XREAL_1:6;
j <= len (p /^ n) by A6, FINSEQ_1:1;
then j <= (len p) - n by A1, RFINSEQ:def_1;
then A8: j + n <= len p by XREAL_1:19;
1 <= 1 + n by NAT_1:11;
then 1 <= j + n by A7, XXREAL_0:2;
then j + n in Seg (len p) by A8, FINSEQ_1:1;
then A9: j + n in dom p by FINSEQ_1:def_3;
A10: i in Seg (len (p /^ n)) by A2, FINSEQ_1:def_3;
then 1 <= i by FINSEQ_1:1;
then A11: 1 + n <= i + n by XREAL_1:6;
i <= len (p /^ n) by A10, FINSEQ_1:1;
then i <= (len p) - n by A1, RFINSEQ:def_1;
then A12: i + n <= len p by XREAL_1:19;
1 <= 1 + n by NAT_1:11;
then 1 <= i + n by A11, XXREAL_0:2;
then i + n in Seg (len p) by A12, FINSEQ_1:1;
then A13: i + n in dom p by FINSEQ_1:def_3;
A14: (p /^ n) . j = p . (j + n) by A1, A3, RFINSEQ:def_1;
(p /^ n) . i = p . (i + n) by A1, A2, RFINSEQ:def_1;
hence (p /^ n) . i < (p /^ n) . j by A14, A13, A9, A5, SEQM_3:def_1; ::_thesis: verum
end;
hence p /^ n is increasing FinSequence of REAL by SEQM_3:def_1; ::_thesis: verum
end;
theorem Th35: :: INTEGRA1:35
for p being increasing FinSequence of REAL
for i, j being Element of NAT st j in dom p & i <= j holds
mid (p,i,j) is increasing FinSequence of REAL
proof
let p be increasing FinSequence of REAL ; ::_thesis: for i, j being Element of NAT st j in dom p & i <= j holds
mid (p,i,j) is increasing FinSequence of REAL
let i, j be Element of NAT ; ::_thesis: ( j in dom p & i <= j implies mid (p,i,j) is increasing FinSequence of REAL )
assume that
A1: j in dom p and
A2: i <= j ; ::_thesis: mid (p,i,j) is increasing FinSequence of REAL
j in Seg (len p) by A1, FINSEQ_1:def_3;
then j <= len p by FINSEQ_1:1;
then i <= len p by A2, XXREAL_0:2;
then p /^ (i -' 1) is increasing FinSequence of REAL by Th34, NAT_D:44;
then A3: (p /^ (i -' 1)) | (Seg ((j -' i) + 1)) is increasing FinSequence of REAL by FINSEQ_1:18, SEQ_4:139;
mid (p,i,j) = (p /^ (i -' 1)) | ((j -' i) + 1) by A2, FINSEQ_6:def_3;
hence mid (p,i,j) is increasing FinSequence of REAL by A3, FINSEQ_1:def_15; ::_thesis: verum
end;
Lm1: for i, j being Element of NAT
for f being FinSequence st i in dom f & j in dom f & i <= j holds
len (mid (f,i,j)) = (j - i) + 1
proof
let i, j be Element of NAT ; ::_thesis: for f being FinSequence st i in dom f & j in dom f & i <= j holds
len (mid (f,i,j)) = (j - i) + 1
let D be FinSequence; ::_thesis: ( i in dom D & j in dom D & i <= j implies len (mid (D,i,j)) = (j - i) + 1 )
assume that
A1: i in dom D and
A2: j in dom D and
A3: i <= j ; ::_thesis: len (mid (D,i,j)) = (j - i) + 1
j in Seg (len D) by A2, FINSEQ_1:def_3;
then j <= len D by FINSEQ_1:1;
then i <= len D by A3, XXREAL_0:2;
then i -' 1 <= len D by NAT_D:44;
then A4: len (D /^ (i -' 1)) = (len D) - (i -' 1) by RFINSEQ:def_1;
reconsider D1 = D /^ (i -' 1) as FinSequence ;
reconsider k = (j -' i) + 1 as Element of NAT ;
i in Seg (len D) by A1, FINSEQ_1:def_3;
then 1 <= i by FINSEQ_1:1;
then j - (i -' 1) = j - (i - 1) by XREAL_1:233;
then A5: j - (i -' 1) = (j - i) + 1 ;
j in Seg (len D) by A2, FINSEQ_1:def_3;
then j <= len D by FINSEQ_1:1;
then j - (i -' 1) <= (len D) - (i -' 1) by XREAL_1:9;
then A6: (j -' i) + 1 <= len (D /^ (i -' 1)) by A3, A4, A5, XREAL_1:233;
mid (D,i,j) = (D /^ (i -' 1)) | ((j -' i) + 1) by A3, FINSEQ_6:def_3
.= D1 | (Seg k) by FINSEQ_1:def_15 ;
then len (mid (D,i,j)) = (j -' i) + 1 by A6, FINSEQ_1:17;
hence len (mid (D,i,j)) = (j - i) + 1 by A3, XREAL_1:233; ::_thesis: verum
end;
theorem Th36: :: INTEGRA1:36
for i, j being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j holds
ex B being non empty closed_interval Subset of REAL st
( lower_bound B = (mid (D,i,j)) . 1 & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B )
proof
let i, j be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j holds
ex B being non empty closed_interval Subset of REAL st
( lower_bound B = (mid (D,i,j)) . 1 & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B )
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D & j in dom D & i <= j holds
ex B being non empty closed_interval Subset of REAL st
( lower_bound B = (mid (D,i,j)) . 1 & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B )
let D be Division of A; ::_thesis: ( i in dom D & j in dom D & i <= j implies ex B being non empty closed_interval Subset of REAL st
( lower_bound B = (mid (D,i,j)) . 1 & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B ) )
assume that
A1: i in dom D and
A2: j in dom D and
A3: i <= j ; ::_thesis: ex B being non empty closed_interval Subset of REAL st
( lower_bound B = (mid (D,i,j)) . 1 & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B )
j in Seg (len D) by A2, FINSEQ_1:def_3;
then j <= len D by FINSEQ_1:1;
then i <= len D by A3, XXREAL_0:2;
then i -' 1 <= len D by NAT_D:44;
then A4: len (D /^ (i -' 1)) = (len D) - (i -' 1) by RFINSEQ:def_1;
reconsider D1 = D /^ (i -' 1) as FinSequence of REAL ;
reconsider k = (j -' i) + 1 as Element of NAT ;
i in Seg (len D) by A1, FINSEQ_1:def_3;
then 1 <= i by FINSEQ_1:1;
then j - (i -' 1) = j - (i - 1) by XREAL_1:233;
then A5: j - (i -' 1) = (j - i) + 1 ;
j in Seg (len D) by A2, FINSEQ_1:def_3;
then j <= len D by FINSEQ_1:1;
then j - (i -' 1) <= (len D) - (i -' 1) by XREAL_1:9;
then A6: (j -' i) + 1 <= len (D /^ (i -' 1)) by A3, A4, A5, XREAL_1:233;
A7: mid (D,i,j) = (D /^ (i -' 1)) | ((j -' i) + 1) by A3, FINSEQ_6:def_3
.= D1 | (Seg k) by FINSEQ_1:def_15 ;
then 0 < len (mid (D,i,j)) by A6, FINSEQ_1:17;
then reconsider M = mid (D,i,j) as non empty increasing FinSequence of REAL by A2, A3, Th35;
(j -' i) + 1 >= 0 + 1 by XREAL_1:6;
then A8: 1 <= len M by A6, A7, FINSEQ_1:17;
then len M in Seg (len M) by FINSEQ_1:1;
then A9: len M in dom M by FINSEQ_1:def_3;
1 in Seg (len M) by A8, FINSEQ_1:1;
then A10: 1 in dom M by FINSEQ_1:def_3;
then M . 1 <= M . (len M) by A8, A9, SEQ_4:137;
then reconsider B = [.(M . 1),(M . (len M)).] as non empty closed_interval Subset of REAL by MEASURE5:14;
A11: B = [.(lower_bound B),(upper_bound B).] by Th4;
then A12: lower_bound B = M . 1 by Th5;
A13: M . (len M) = upper_bound B by A11, Th5;
for x being Real st x in rng M holds
x in B
proof
let x be Real; ::_thesis: ( x in rng M implies x in B )
assume x in rng M ; ::_thesis: x in B
then consider i being Element of NAT such that
A14: i in dom M and
A15: x = M . i by PARTFUN1:3;
A16: i in Seg (len M) by A14, FINSEQ_1:def_3;
then i <= len M by FINSEQ_1:1;
then A17: M . i <= M . (len M) by A9, A14, SEQ_4:137;
1 <= i by A16, FINSEQ_1:1;
then M . 1 <= M . i by A10, A14, SEQ_4:137;
then M . i in { a where a is Real : ( M . 1 <= a & a <= M . (len M) ) } by A17;
hence x in B by A15, RCOMP_1:def_1; ::_thesis: verum
end;
then rng M c= B by SUBSET_1:2;
then M is Division of B by A13, Def2;
hence ex B being non empty closed_interval Subset of REAL st
( lower_bound B = (mid (D,i,j)) . 1 & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B ) by A12, A13; ::_thesis: verum
end;
theorem Th37: :: INTEGRA1:37
for i, j being Element of NAT
for A, B being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j & D . i >= lower_bound B & D . j = upper_bound B holds
mid (D,i,j) is Division of B
proof
let i, j be Element of NAT ; ::_thesis: for A, B being non empty closed_interval Subset of REAL
for D being Division of A st i in dom D & j in dom D & i <= j & D . i >= lower_bound B & D . j = upper_bound B holds
mid (D,i,j) is Division of B
let A, B be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A st i in dom D & j in dom D & i <= j & D . i >= lower_bound B & D . j = upper_bound B holds
mid (D,i,j) is Division of B
let D be Division of A; ::_thesis: ( i in dom D & j in dom D & i <= j & D . i >= lower_bound B & D . j = upper_bound B implies mid (D,i,j) is Division of B )
assume that
A1: i in dom D and
A2: j in dom D and
A3: i <= j and
A4: D . i >= lower_bound B and
A5: D . j = upper_bound B ; ::_thesis: mid (D,i,j) is Division of B
A6: (((j - i) + 1) + i) - 1 = j ;
i in Seg (len D) by A1, FINSEQ_1:def_3;
then A7: 1 <= i by FINSEQ_1:1;
0 <= j - i by A3, XREAL_1:48;
then A8: 0 + 1 <= (j - i) + 1 by XREAL_1:6;
j in Seg (len D) by A2, FINSEQ_1:def_3;
then A9: j <= len D by FINSEQ_1:1;
consider A1 being non empty closed_interval Subset of REAL such that
A10: lower_bound A1 = (mid (D,i,j)) . 1 and
A11: upper_bound A1 = (mid (D,i,j)) . (len (mid (D,i,j))) and
A12: mid (D,i,j) is Division of A1 by A1, A2, A3, Th36;
A13: len (mid (D,i,j)) = (j - i) + 1 by A1, A2, A3, Lm1;
A14: (1 + i) - 1 = i ;
for x being Real st x in A1 holds
x in B
proof
let x be Real; ::_thesis: ( x in A1 implies x in B )
assume x in A1 ; ::_thesis: x in B
then x in [.(lower_bound A1),(upper_bound A1).] by Th4;
then x in { a where a is Real : ( lower_bound A1 <= a & a <= upper_bound A1 ) } by RCOMP_1:def_1;
then A15: ex a being Real st
( x = a & lower_bound A1 <= a & a <= upper_bound A1 ) ;
then D . i <= x by A3, A10, A7, A9, A8, A14, FINSEQ_6:122;
then A16: lower_bound B <= x by A4, XXREAL_0:2;
x <= upper_bound B by A3, A5, A11, A13, A7, A9, A8, A6, A15, FINSEQ_6:122;
then x in { a where a is Real : ( lower_bound B <= a & a <= upper_bound B ) } by A16;
then x in [.(lower_bound B),(upper_bound B).] by RCOMP_1:def_1;
hence x in B by Th4; ::_thesis: verum
end;
then A17: A1 c= B by SUBSET_1:2;
rng (mid (D,i,j)) c= A1 by A12, Def2;
then A18: rng (mid (D,i,j)) c= B by A17, XBOOLE_1:1;
(mid (D,i,j)) . (len (mid (D,i,j))) = D . j by A3, A13, A7, A9, A8, A6, FINSEQ_6:122;
hence mid (D,i,j) is Division of B by A5, A12, A18, Def2; ::_thesis: verum
end;
definition
let p be FinSequence of REAL ;
func PartSums p -> FinSequence of REAL means :Def20: :: INTEGRA1:def 20
( len it = len p & ( for i being Nat st i in dom p holds
it . i = Sum (p | i) ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = len p & ( for i being Nat st i in dom p holds
b1 . i = Sum (p | i) ) )
proof
deffunc H1( Nat) -> Element of REAL = Sum (p | $1);
consider IT being FinSequence of REAL such that
A1: ( len IT = len p & ( for i being Nat st i in dom IT holds
IT . i = H1(i) ) ) from FINSEQ_2:sch_1();
take IT ; ::_thesis: ( len IT = len p & ( for i being Nat st i in dom p holds
IT . i = Sum (p | i) ) )
thus len IT = len p by A1; ::_thesis: for i being Nat st i in dom p holds
IT . i = Sum (p | i)
let i be Nat; ::_thesis: ( i in dom p implies IT . i = Sum (p | i) )
assume i in dom p ; ::_thesis: IT . i = Sum (p | i)
then i in dom IT by A1, FINSEQ_3:29;
hence IT . i = Sum (p | i) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being FinSequence of REAL st len b1 = len p & ( for i being Nat st i in dom p holds
b1 . i = Sum (p | i) ) & len b2 = len p & ( for i being Nat st i in dom p holds
b2 . i = Sum (p | i) ) holds
b1 = b2
proof
let p1, p2 be FinSequence of REAL ; ::_thesis: ( len p1 = len p & ( for i being Nat st i in dom p holds
p1 . i = Sum (p | i) ) & len p2 = len p & ( for i being Nat st i in dom p holds
p2 . i = Sum (p | i) ) implies p1 = p2 )
assume that
A2: len p1 = len p and
A3: for i being Nat st i in dom p holds
p1 . i = Sum (p | i) and
A4: len p2 = len p and
A5: for i being Nat st i in dom p holds
p2 . i = Sum (p | i) ; ::_thesis: p1 = p2
for i being Nat st 1 <= i & i <= len p1 holds
p1 . i = p2 . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len p1 implies p1 . i = p2 . i )
assume that
A6: 1 <= i and
A7: i <= len p1 ; ::_thesis: p1 . i = p2 . i
A8: i in dom p by A2, A6, A7, FINSEQ_3:25;
then p1 . i = Sum (p | i) by A3;
hence p1 . i = p2 . i by A5, A8; ::_thesis: verum
end;
hence p1 = p2 by A2, A4, FINSEQ_1:14; ::_thesis: verum
end;
end;
:: deftheorem Def20 defines PartSums INTEGRA1:def_20_:_
for p, b2 being FinSequence of REAL holds
( b2 = PartSums p iff ( len b2 = len p & ( for i being Nat st i in dom p holds
b2 . i = Sum (p | i) ) ) );
theorem Th38: :: INTEGRA1:38
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
for i being non empty Element of NAT st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i)))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
for i being non empty Element of NAT st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i)))
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
for i being non empty Element of NAT st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i)))
let f be Function of A,REAL; ::_thesis: ( D1 <= D2 & f | A is bounded_above implies for i being non empty Element of NAT st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i))) )
assume that
A1: D1 <= D2 and
A2: f | A is bounded_above ; ::_thesis: for i being non empty Element of NAT st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i)))
for i being non empty Nat st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i)))
proof
defpred S1[ Nat] means ( $1 in dom D1 implies Sum ((upper_volume (f,D1)) | $1) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,$1))) );
A3: S1[1]
proof
reconsider g = f | (divset (D1,1)) as PartFunc of (divset (D1,1)),REAL by PARTFUN1:10;
set B = divset (D1,1);
set DD1 = mid (D1,1,1);
A4: dom g = (dom f) /\ (divset (D1,1)) by RELAT_1:61;
assume A5: 1 in dom D1 ; ::_thesis: Sum ((upper_volume (f,D1)) | 1) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,1)))
then A6: D1 . 1 = upper_bound (divset (D1,1)) by Def4;
then A7: D2 . (indx (D2,D1,1)) = upper_bound (divset (D1,1)) by A1, A5, Def19;
D1 . 1 >= lower_bound (divset (D1,1)) by A6, SEQ_4:11;
then reconsider DD1 = mid (D1,1,1) as Division of divset (D1,1) by A5, A6, Th37;
1 in Seg (len D1) by A5, FINSEQ_1:def_3;
then A8: 1 <= len D1 by FINSEQ_1:1;
then A9: len (mid (D1,1,1)) = (1 -' 1) + 1 by FINSEQ_6:118;
A10: len (upper_volume (g,DD1)) = len DD1 by Def6
.= 1 by A9, XREAL_1:235 ;
A11: len (mid (D1,1,1)) = 1 by A9, XREAL_1:235;
then A12: len (mid (D1,1,1)) = len (D1 | 1) by A8, FINSEQ_1:59;
for k being Nat st 1 <= k & k <= len (mid (D1,1,1)) holds
(mid (D1,1,1)) . k = (D1 | 1) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (mid (D1,1,1)) implies (mid (D1,1,1)) . k = (D1 | 1) . k )
assume that
A13: 1 <= k and
A14: k <= len (mid (D1,1,1)) ; ::_thesis: (mid (D1,1,1)) . k = (D1 | 1) . k
k in Seg (len (D1 | 1)) by A12, A13, A14, FINSEQ_1:1;
then k in dom (D1 | 1) by FINSEQ_1:def_3;
then k in dom (D1 | (Seg 1)) by FINSEQ_1:def_15;
then A15: (D1 | (Seg 1)) . k = D1 . k by FUNCT_1:47;
A16: k = 1 by A11, A13, A14, XXREAL_0:1;
then (mid (D1,1,1)) . k = D1 . ((1 + 1) - 1) by A8, FINSEQ_6:118;
hence (mid (D1,1,1)) . k = (D1 | 1) . k by A16, A15, FINSEQ_1:def_15; ::_thesis: verum
end;
then A17: mid (D1,1,1) = D1 | 1 by A12, FINSEQ_1:14;
A18: for i being Nat st 1 <= i & i <= len (upper_volume (g,DD1)) holds
(upper_volume (g,DD1)) . i = ((upper_volume (f,D1)) | 1) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len (upper_volume (g,DD1)) implies (upper_volume (g,DD1)) . i = ((upper_volume (f,D1)) | 1) . i )
assume that
A19: 1 <= i and
A20: i <= len (upper_volume (g,DD1)) ; ::_thesis: (upper_volume (g,DD1)) . i = ((upper_volume (f,D1)) | 1) . i
A21: 1 in Seg 1 by FINSEQ_1:3;
dom (D1 | (Seg 1)) = (dom D1) /\ (Seg 1) by RELAT_1:61;
then A22: 1 in dom (D1 | (Seg 1)) by A5, A21, XBOOLE_0:def_4;
dom (upper_volume (f,D1)) = Seg (len (upper_volume (f,D1))) by FINSEQ_1:def_3
.= Seg (len D1) by Def6 ;
then A23: dom ((upper_volume (f,D1)) | (Seg 1)) = (Seg (len D1)) /\ (Seg 1) by RELAT_1:61
.= Seg 1 by A8, FINSEQ_1:7 ;
len DD1 = 1 by A9, XREAL_1:235;
then A24: 1 in dom DD1 by A21, FINSEQ_1:def_3;
A25: ((upper_volume (f,D1)) | 1) . i = ((upper_volume (f,D1)) | (Seg 1)) . i by FINSEQ_1:def_15
.= ((upper_volume (f,D1)) | (Seg 1)) . 1 by A10, A19, A20, XXREAL_0:1
.= (upper_volume (f,D1)) . 1 by A23, FINSEQ_1:3, FUNCT_1:47
.= (upper_bound (rng (f | (divset (D1,1))))) * (vol (divset (D1,1))) by A5, Def6 ;
A26: divset (D1,1) = [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by Th4
.= [.(lower_bound A),(upper_bound (divset (D1,1))).] by A5, Def4
.= [.(lower_bound A),(D1 . 1).] by A5, Def4 ;
A27: (upper_volume (g,DD1)) . i = (upper_volume (g,DD1)) . 1 by A10, A19, A20, XXREAL_0:1
.= (upper_bound (rng (g | (divset (DD1,1))))) * (vol (divset (DD1,1))) by A24, Def6 ;
divset (DD1,1) = [.(lower_bound (divset (DD1,1))),(upper_bound (divset (DD1,1))).] by Th4
.= [.(lower_bound (divset (D1,1))),(upper_bound (divset (DD1,1))).] by A24, Def4
.= [.(lower_bound (divset (D1,1))),(DD1 . 1).] by A24, Def4
.= [.(lower_bound A),((D1 | 1) . 1).] by A5, A17, Def4
.= [.(lower_bound A),((D1 | (Seg 1)) . 1).] by FINSEQ_1:def_15
.= [.(lower_bound A),(D1 . 1).] by A22, FUNCT_1:47 ;
hence (upper_volume (g,DD1)) . i = ((upper_volume (f,D1)) | 1) . i by A4, A27, A26, A25, RELAT_1:68; ::_thesis: verum
end;
A28: g | (divset (D1,1)) is bounded_above
proof
consider a being real number such that
A29: for x being set st x in A /\ (dom f) holds
f . x <= a by A2, RFUNCT_1:70;
for x being set st x in (divset (D1,1)) /\ (dom g) holds
g . x <= a
proof
let x be set ; ::_thesis: ( x in (divset (D1,1)) /\ (dom g) implies g . x <= a )
A30: dom g c= dom f by RELAT_1:60;
assume x in (divset (D1,1)) /\ (dom g) ; ::_thesis: g . x <= a
then A31: x in dom g by XBOOLE_0:def_4;
A32: A /\ (dom f) = dom f by XBOOLE_1:28;
then x in A /\ (dom f) by A31, A30;
then reconsider x = x as Element of A ;
f . x <= a by A29, A31, A32, A30;
hence g . x <= a by A31, FUNCT_1:47; ::_thesis: verum
end;
hence g | (divset (D1,1)) is bounded_above by RFUNCT_1:70; ::_thesis: verum
end;
A33: rng D2 c= A by Def2;
A34: indx (D2,D1,1) in dom D2 by A1, A5, Def19;
then A35: indx (D2,D1,1) in Seg (len D2) by FINSEQ_1:def_3;
then A36: 1 <= indx (D2,D1,1) by FINSEQ_1:1;
A37: indx (D2,D1,1) <= len D2 by A35, FINSEQ_1:1;
then 1 <= len D2 by A36, XXREAL_0:2;
then 1 in Seg (len D2) by FINSEQ_1:1;
then A38: 1 in dom D2 by FINSEQ_1:def_3;
then D2 . 1 in rng D2 by FUNCT_1:def_3;
then D2 . 1 in A by A33;
then D2 . 1 in [.(lower_bound A),(upper_bound A).] by Th4;
then D2 . 1 in { a where a is Real : ( lower_bound A <= a & a <= upper_bound A ) } by RCOMP_1:def_1;
then ex a being Real st
( D2 . 1 = a & lower_bound A <= a & a <= upper_bound A ) ;
then D2 . 1 >= lower_bound (divset (D1,1)) by A5, Def4;
then reconsider DD2 = mid (D2,1,(indx (D2,D1,1))) as Division of divset (D1,1) by A34, A36, A38, A7, Th37;
indx (D2,D1,1) in dom D2 by A1, A5, Def19;
then A39: indx (D2,D1,1) in Seg (len D2) by FINSEQ_1:def_3;
then A40: 1 <= indx (D2,D1,1) by FINSEQ_1:1;
A41: indx (D2,D1,1) <= len D2 by A39, FINSEQ_1:1;
then A42: 1 <= len D2 by A40, XXREAL_0:2;
then len (mid (D2,1,(indx (D2,D1,1)))) = ((indx (D2,D1,1)) -' 1) + 1 by A40, A41, FINSEQ_6:118;
then A43: len (mid (D2,1,(indx (D2,D1,1)))) = ((indx (D2,D1,1)) - 1) + 1 by A40, XREAL_1:233;
then A44: len (mid (D2,1,(indx (D2,D1,1)))) = len (D2 | (indx (D2,D1,1))) by A41, FINSEQ_1:59;
A45: for k being Nat st 1 <= k & k <= len (mid (D2,1,(indx (D2,D1,1)))) holds
(mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (mid (D2,1,(indx (D2,D1,1)))) implies (mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k )
assume that
A46: 1 <= k and
A47: k <= len (mid (D2,1,(indx (D2,D1,1)))) ; ::_thesis: (mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k
k in Seg (len (D2 | (indx (D2,D1,1)))) by A44, A46, A47, FINSEQ_1:1;
then k in dom (D2 | (indx (D2,D1,1))) by FINSEQ_1:def_3;
then k in dom (D2 | (Seg (indx (D2,D1,1)))) by FINSEQ_1:def_15;
then A48: (D2 | (Seg (indx (D2,D1,1)))) . k = D2 . k by FUNCT_1:47;
k in NAT by ORDINAL1:def_12;
then (mid (D2,1,(indx (D2,D1,1)))) . k = D2 . ((k + 1) -' 1) by A40, A41, A42, A46, A47, FINSEQ_6:118;
then (mid (D2,1,(indx (D2,D1,1)))) . k = D2 . ((k + 1) - 1) by NAT_1:11, XREAL_1:233;
hence (mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k by A48, FINSEQ_1:def_15; ::_thesis: verum
end;
then A49: mid (D2,1,(indx (D2,D1,1))) = D2 | (indx (D2,D1,1)) by A44, FINSEQ_1:14;
A50: for i being Nat st 1 <= i & i <= len (upper_volume (g,DD2)) holds
(upper_volume (g,DD2)) . i = ((upper_volume (f,D2)) | (indx (D2,D1,1))) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len (upper_volume (g,DD2)) implies (upper_volume (g,DD2)) . i = ((upper_volume (f,D2)) | (indx (D2,D1,1))) . i )
assume that
A51: 1 <= i and
A52: i <= len (upper_volume (g,DD2)) ; ::_thesis: (upper_volume (g,DD2)) . i = ((upper_volume (f,D2)) | (indx (D2,D1,1))) . i
A53: i <= len DD2 by A52, Def6;
then A54: i in Seg (len DD2) by A51, FINSEQ_1:1;
then A55: i in dom DD2 by FINSEQ_1:def_3;
divset (DD2,i) = divset (D2,i)
proof
Seg (indx (D2,D1,1)) c= Seg (len D2) by A41, FINSEQ_1:5;
then i in Seg (len D2) by A43, A54;
then A56: i in dom D2 by FINSEQ_1:def_3;
now__::_thesis:_divset_(DD2,i)_=_divset_(D2,i)
percases ( i = 1 or i <> 1 ) ;
supposeA57: i = 1 ; ::_thesis: divset (DD2,i) = divset (D2,i)
then A58: 1 in dom (D2 | (Seg (indx (D2,D1,1)))) by A49, A55, FINSEQ_1:def_15;
then 1 in (dom D2) /\ (Seg (indx (D2,D1,1))) by RELAT_1:61;
then A59: 1 in dom D2 by XBOOLE_0:def_4;
A60: divset (D2,i) = [.(lower_bound (divset (D2,1))),(upper_bound (divset (D2,1))).] by A57, Th4
.= [.(lower_bound A),(upper_bound (divset (D2,1))).] by A59, Def4
.= [.(lower_bound A),(D2 . 1).] by A59, Def4 ;
divset (DD2,i) = [.(lower_bound (divset (DD2,1))),(upper_bound (divset (DD2,1))).] by A57, Th4
.= [.(lower_bound (divset (D1,1))),(upper_bound (divset (DD2,1))).] by A55, A57, Def4
.= [.(lower_bound (divset (D1,1))),(DD2 . 1).] by A55, A57, Def4
.= [.(lower_bound (divset (D1,1))),((D2 | (indx (D2,D1,1))) . 1).] by A45, A53, A57
.= [.(lower_bound (divset (D1,1))),((D2 | (Seg (indx (D2,D1,1)))) . 1).] by FINSEQ_1:def_15
.= [.(lower_bound (divset (D1,1))),(D2 . 1).] by A58, FUNCT_1:47
.= [.(lower_bound A),(D2 . 1).] by A5, Def4 ;
hence divset (DD2,i) = divset (D2,i) by A60; ::_thesis: verum
end;
supposeA61: i <> 1 ; ::_thesis: divset (DD2,i) = divset (D2,i)
A62: i - 1 in dom (D2 | (Seg (indx (D2,D1,1))))
proof
not i is trivial by A51, A61, NAT_2:def_1;
then A63: i >= 1 + 1 by NAT_2:29;
then A64: 1 <= i - 1 by XREAL_1:19;
A65: ex j being Nat st i = 1 + j by A51, NAT_1:10;
A66: i - 1 <= (indx (D2,D1,1)) - 0 by A43, A53, XREAL_1:13;
then i - 1 <= len D2 by A37, XXREAL_0:2;
then i - 1 in Seg (len D2) by A65, A64, FINSEQ_1:1;
then A67: i - 1 in dom D2 by FINSEQ_1:def_3;
i - 1 >= 1 by A63, XREAL_1:19;
then i - 1 in Seg (indx (D2,D1,1)) by A65, A66, FINSEQ_1:1;
then i - 1 in (dom D2) /\ (Seg (indx (D2,D1,1))) by A67, XBOOLE_0:def_4;
hence i - 1 in dom (D2 | (Seg (indx (D2,D1,1)))) by RELAT_1:61; ::_thesis: verum
end;
DD2 . (i - 1) = (D2 | (indx (D2,D1,1))) . (i - 1) by A44, A45, FINSEQ_1:14
.= (D2 | (Seg (indx (D2,D1,1)))) . (i - 1) by FINSEQ_1:def_15 ;
then A68: DD2 . (i - 1) = D2 . (i - 1) by A62, FUNCT_1:47;
i <= len D2 by A43, A37, A53, XXREAL_0:2;
then i in Seg (len D2) by A51, FINSEQ_1:1;
then i in dom D2 by FINSEQ_1:def_3;
then i in (dom D2) /\ (Seg (indx (D2,D1,1))) by A43, A54, XBOOLE_0:def_4;
then A69: i in dom (D2 | (Seg (indx (D2,D1,1)))) by RELAT_1:61;
DD2 . i = (D2 | (indx (D2,D1,1))) . i by A44, A45, FINSEQ_1:14
.= (D2 | (Seg (indx (D2,D1,1)))) . i by FINSEQ_1:def_15 ;
then A70: DD2 . i = D2 . i by A69, FUNCT_1:47;
A71: divset (D2,i) = [.(lower_bound (divset (D2,i))),(upper_bound (divset (D2,i))).] by Th4
.= [.(D2 . (i - 1)),(upper_bound (divset (D2,i))).] by A56, A61, Def4
.= [.(D2 . (i - 1)),(D2 . i).] by A56, A61, Def4 ;
divset (DD2,i) = [.(lower_bound (divset (DD2,i))),(upper_bound (divset (DD2,i))).] by Th4
.= [.(DD2 . (i - 1)),(upper_bound (divset (DD2,i))).] by A55, A61, Def4
.= [.(D2 . (i - 1)),(D2 . i).] by A55, A61, A68, A70, Def4 ;
hence divset (DD2,i) = divset (D2,i) by A71; ::_thesis: verum
end;
end;
end;
hence divset (DD2,i) = divset (D2,i) ; ::_thesis: verum
end;
then A72: (upper_volume (g,DD2)) . i = (upper_bound (rng (g | (divset (D2,i))))) * (vol (divset (D2,i))) by A55, Def6;
Seg (indx (D2,D1,1)) c= Seg (len D2) by A41, FINSEQ_1:5;
then i in Seg (len D2) by A43, A54;
then A73: i in dom D2 by FINSEQ_1:def_3;
A74: i in dom DD2 by A54, FINSEQ_1:def_3;
A75: now__::_thesis:_(_lower_bound_(divset_(D2,i))_in_[.(lower_bound_(divset_(D1,1))),(upper_bound_(divset_(D1,1))).]_&_upper_bound_(divset_(D2,i))_in_[.(lower_bound_(divset_(D1,1))),(upper_bound_(divset_(D1,1))).]_)
percases ( i = 1 or i <> 1 ) ;
supposeA76: i = 1 ; ::_thesis: ( lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] & upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] )
then 1 in dom (D2 | (Seg (indx (D2,D1,1)))) by A49, A74, FINSEQ_1:def_15;
then 1 in (dom D2) /\ (Seg (indx (D2,D1,1))) by RELAT_1:61;
then A77: 1 in dom D2 by XBOOLE_0:def_4;
then A78: D2 . 1 <= D2 . (indx (D2,D1,1)) by A34, A36, SEQ_4:137;
lower_bound (divset (D2,i)) = lower_bound A by A76, A77, Def4;
then A79: lower_bound (divset (D2,i)) = lower_bound (divset (D1,1)) by A5, Def4;
upper_bound (divset (D2,i)) = D2 . 1 by A76, A77, Def4;
then upper_bound (divset (D2,i)) <= D1 . 1 by A1, A5, A78, Def19;
then A80: upper_bound (divset (D2,i)) <= upper_bound (divset (D1,1)) by A5, Def4;
lower_bound (divset (D1,1)) <= upper_bound (divset (D1,1)) by SEQ_4:11;
hence lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by A79, XXREAL_1:1; ::_thesis: upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).]
lower_bound (divset (D2,i)) <= upper_bound (divset (D2,i)) by SEQ_4:11;
then upper_bound (divset (D2,i)) in { r where r is Real : ( lower_bound (divset (D1,1)) <= r & r <= upper_bound (divset (D1,1)) ) } by A79, A80;
hence upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by RCOMP_1:def_1; ::_thesis: verum
end;
supposeA81: i <> 1 ; ::_thesis: ( lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] & upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] )
then not i is trivial by A51, NAT_2:def_1;
then i >= 1 + 1 by NAT_2:29;
then A82: 1 <= i - 1 by XREAL_1:19;
A83: ex j being Nat st i = 1 + j by A51, NAT_1:10;
A84: rng D2 c= A by Def2;
A85: lower_bound (divset (D2,i)) = D2 . (i - 1) by A73, A81, Def4;
A86: lower_bound (divset (D1,1)) = lower_bound A by A5, Def4;
A87: i - 1 <= (indx (D2,D1,1)) - 0 by A43, A53, XREAL_1:13;
then i - 1 <= len D2 by A37, XXREAL_0:2;
then i - 1 in Seg (len D2) by A83, A82, FINSEQ_1:1;
then A88: i - 1 in dom D2 by FINSEQ_1:def_3;
then D2 . (i - 1) in rng D2 by FUNCT_1:def_3;
then A89: lower_bound (divset (D2,i)) >= lower_bound (divset (D1,1)) by A85, A86, A84, SEQ_4:def_2;
A90: upper_bound (divset (D1,1)) = D1 . 1 by A5, Def4;
D2 . (i - 1) <= D2 . (indx (D2,D1,1)) by A34, A87, A88, SEQ_4:137;
then lower_bound (divset (D2,i)) <= upper_bound (divset (D1,1)) by A1, A5, A85, A90, Def19;
then lower_bound (divset (D2,i)) in { r where r is Real : ( lower_bound (divset (D1,1)) <= r & r <= upper_bound (divset (D1,1)) ) } by A89;
hence lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by RCOMP_1:def_1; ::_thesis: upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).]
A91: upper_bound (divset (D2,i)) = D2 . i by A73, A81, Def4;
D2 . i in rng D2 by A73, FUNCT_1:def_3;
then A92: upper_bound (divset (D2,i)) >= lower_bound (divset (D1,1)) by A91, A86, A84, SEQ_4:def_2;
D2 . i <= D2 . (indx (D2,D1,1)) by A43, A34, A53, A73, SEQ_4:137;
then upper_bound (divset (D2,i)) <= upper_bound (divset (D1,1)) by A1, A5, A91, A90, Def19;
then upper_bound (divset (D2,i)) in { r where r is Real : ( lower_bound (divset (D1,1)) <= r & r <= upper_bound (divset (D1,1)) ) } by A92;
hence upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by RCOMP_1:def_1; ::_thesis: verum
end;
end;
end;
A93: divset (D1,1) = [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by Th4;
A94: Seg (indx (D2,D1,1)) c= Seg (len D2) by A41, FINSEQ_1:5;
then i in Seg (len D2) by A43, A54;
then A95: i in dom D2 by FINSEQ_1:def_3;
divset (D2,i) = [.(lower_bound (divset (D2,i))),(upper_bound (divset (D2,i))).] by Th4;
then A96: divset (D2,i) c= divset (D1,1) by A93, A75, XXREAL_2:def_12;
A97: dom ((upper_volume (f,D2)) | (Seg (indx (D2,D1,1)))) = (dom (upper_volume (f,D2))) /\ (Seg (indx (D2,D1,1))) by RELAT_1:61
.= (Seg (len (upper_volume (f,D2)))) /\ (Seg (indx (D2,D1,1))) by FINSEQ_1:def_3
.= (Seg (len D2)) /\ (Seg (indx (D2,D1,1))) by Def6
.= Seg (indx (D2,D1,1)) by A94, XBOOLE_1:28 ;
((upper_volume (f,D2)) | (indx (D2,D1,1))) . i = ((upper_volume (f,D2)) | (Seg (indx (D2,D1,1)))) . i by FINSEQ_1:def_15
.= (upper_volume (f,D2)) . i by A43, A54, A97, FUNCT_1:47
.= (upper_bound (rng (f | (divset (D2,i))))) * (vol (divset (D2,i))) by A95, Def6 ;
hence (upper_volume (g,DD2)) . i = ((upper_volume (f,D2)) | (indx (D2,D1,1))) . i by A72, A96, FUNCT_1:51; ::_thesis: verum
end;
1 <= len (upper_volume (f,D1)) by A8, Def6;
then len (upper_volume (g,DD1)) = len ((upper_volume (f,D1)) | 1) by A10, FINSEQ_1:59;
then A98: upper_volume (g,DD1) = (upper_volume (f,D1)) | 1 by A18, FINSEQ_1:14;
A99: indx (D2,D1,1) <= len (upper_volume (f,D2)) by A41, Def6;
len (upper_volume (g,DD2)) = indx (D2,D1,1) by A43, Def6;
then A100: len (upper_volume (g,DD2)) = len ((upper_volume (f,D2)) | (indx (D2,D1,1))) by A99, FINSEQ_1:59;
dom g = A /\ (divset (D1,1)) by A4, FUNCT_2:def_1;
then dom g = divset (D1,1) by A5, Th8, XBOOLE_1:28;
then g is total by PARTFUN1:def_2;
then upper_sum (g,DD1) >= upper_sum (g,DD2) by A11, A28, Th30;
hence Sum ((upper_volume (f,D1)) | 1) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,1))) by A98, A100, A50, FINSEQ_1:14; ::_thesis: verum
end;
A101: for k being non empty Nat st S1[k] holds
S1[k + 1]
proof
let k be non empty Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A102: ( k in dom D1 implies Sum ((upper_volume (f,D1)) | k) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,k))) ) ; ::_thesis: S1[k + 1]
assume A103: k + 1 in dom D1 ; ::_thesis: Sum ((upper_volume (f,D1)) | (k + 1)) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1))))
then A104: k + 1 in Seg (len D1) by FINSEQ_1:def_3;
then A105: 1 <= k + 1 by FINSEQ_1:1;
A106: k + 1 <= len D1 by A104, FINSEQ_1:1;
now__::_thesis:_Sum_((upper_volume_(f,D1))_|_(k_+_1))_>=_Sum_((upper_volume_(f,D2))_|_(indx_(D2,D1,(k_+_1))))
percases ( 1 = k + 1 or 1 <> k + 1 ) ;
suppose 1 = k + 1 ; ::_thesis: Sum ((upper_volume (f,D1)) | (k + 1)) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1))))
hence Sum ((upper_volume (f,D1)) | (k + 1)) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A3, A103; ::_thesis: verum
end;
supposeA107: 1 <> k + 1 ; ::_thesis: Sum ((upper_volume (f,D1)) | (k + 1)) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1))))
set IDK = indx (D2,D1,k);
set IDK1 = indx (D2,D1,(k + 1));
set K1D2 = (upper_volume (f,D2)) | (indx (D2,D1,(k + 1)));
set KD1 = (upper_volume (f,D1)) | k;
set K1D1 = (upper_volume (f,D1)) | (k + 1);
set n = k + 1;
A108: k + 1 <= len (upper_volume (f,D1)) by A106, Def6;
then A109: len ((upper_volume (f,D1)) | (k + 1)) = k + 1 by FINSEQ_1:59;
then consider p1, q1 being FinSequence of REAL such that
A110: len p1 = k and
A111: len q1 = 1 and
A112: (upper_volume (f,D1)) | (k + 1) = p1 ^ q1 by FINSEQ_2:23;
A113: k <= k + 1 by NAT_1:11;
then A114: k <= len D1 by A106, XXREAL_0:2;
then A115: k <= len (upper_volume (f,D1)) by Def6;
then A116: len p1 = len ((upper_volume (f,D1)) | k) by A110, FINSEQ_1:59;
for i being Nat st 1 <= i & i <= len p1 holds
p1 . i = ((upper_volume (f,D1)) | k) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len p1 implies p1 . i = ((upper_volume (f,D1)) | k) . i )
assume that
A117: 1 <= i and
A118: i <= len p1 ; ::_thesis: p1 . i = ((upper_volume (f,D1)) | k) . i
A119: i in Seg (len p1) by A117, A118, FINSEQ_1:1;
then A120: i in dom ((upper_volume (f,D1)) | k) by A116, FINSEQ_1:def_3;
then A121: i in dom ((upper_volume (f,D1)) | (Seg k)) by FINSEQ_1:def_15;
k <= k + 1 by NAT_1:11;
then A122: Seg k c= Seg (k + 1) by FINSEQ_1:5;
A123: dom ((upper_volume (f,D1)) | (k + 1)) = Seg (len ((upper_volume (f,D1)) | (k + 1))) by FINSEQ_1:def_3
.= Seg (k + 1) by A108, FINSEQ_1:59 ;
dom ((upper_volume (f,D1)) | k) = Seg (len ((upper_volume (f,D1)) | k)) by FINSEQ_1:def_3
.= Seg k by A115, FINSEQ_1:59 ;
then i in dom ((upper_volume (f,D1)) | (k + 1)) by A120, A122, A123;
then A124: i in dom ((upper_volume (f,D1)) | (Seg (k + 1))) by FINSEQ_1:def_15;
A125: ((upper_volume (f,D1)) | (k + 1)) . i = ((upper_volume (f,D1)) | (Seg (k + 1))) . i by FINSEQ_1:def_15
.= (upper_volume (f,D1)) . i by A124, FUNCT_1:47 ;
A126: ((upper_volume (f,D1)) | k) . i = ((upper_volume (f,D1)) | (Seg k)) . i by FINSEQ_1:def_15
.= (upper_volume (f,D1)) . i by A121, FUNCT_1:47 ;
i in dom p1 by A119, FINSEQ_1:def_3;
hence p1 . i = ((upper_volume (f,D1)) | k) . i by A112, A126, A125, FINSEQ_1:def_7; ::_thesis: verum
end;
then A127: p1 = (upper_volume (f,D1)) | k by A116, FINSEQ_1:14;
A128: indx (D2,D1,(k + 1)) in dom D2 by A1, A103, Def19;
then A129: indx (D2,D1,(k + 1)) in Seg (len D2) by FINSEQ_1:def_3;
then A130: 1 <= indx (D2,D1,(k + 1)) by FINSEQ_1:1;
not k + 1 is trivial by A107, NAT_2:def_1;
then k + 1 >= 2 by NAT_2:29;
then k >= (1 + 1) - 1 by XREAL_1:20;
then A131: k in Seg (len D1) by A114, FINSEQ_1:1;
then A132: k in dom D1 by FINSEQ_1:def_3;
then A133: indx (D2,D1,k) in dom D2 by A1, Def19;
A134: indx (D2,D1,k) < indx (D2,D1,(k + 1))
proof
k < k + 1 by NAT_1:13;
then A135: D1 . k < D1 . (k + 1) by A103, A132, SEQM_3:def_1;
assume indx (D2,D1,k) >= indx (D2,D1,(k + 1)) ; ::_thesis: contradiction
then A136: D2 . (indx (D2,D1,k)) >= D2 . (indx (D2,D1,(k + 1))) by A133, A128, SEQ_4:137;
D1 . k = D2 . (indx (D2,D1,k)) by A1, A132, Def19;
hence contradiction by A1, A103, A136, A135, Def19; ::_thesis: verum
end;
A137: indx (D2,D1,(k + 1)) >= indx (D2,D1,k)
proof
assume indx (D2,D1,(k + 1)) < indx (D2,D1,k) ; ::_thesis: contradiction
then A138: D2 . (indx (D2,D1,(k + 1))) < D2 . (indx (D2,D1,k)) by A133, A128, SEQM_3:def_1;
D1 . (k + 1) = D2 . (indx (D2,D1,(k + 1))) by A1, A103, Def19;
then D1 . (k + 1) < D1 . k by A1, A132, A138, Def19;
hence contradiction by A103, A132, NAT_1:11, SEQ_4:137; ::_thesis: verum
end;
then consider ID being Nat such that
A139: indx (D2,D1,(k + 1)) = (indx (D2,D1,k)) + ID by NAT_1:10;
reconsider ID = ID as Element of NAT by ORDINAL1:def_12;
A140: len (upper_volume (f,D2)) = len D2 by Def6;
then A141: indx (D2,D1,(k + 1)) <= len (upper_volume (f,D2)) by A129, FINSEQ_1:1;
then len ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = (indx (D2,D1,k)) + ((indx (D2,D1,(k + 1))) - (indx (D2,D1,k))) by FINSEQ_1:59;
then consider p2, q2 being FinSequence of REAL such that
A142: len p2 = indx (D2,D1,k) and
A143: len q2 = (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) and
A144: (upper_volume (f,D2)) | (indx (D2,D1,(k + 1))) = p2 ^ q2 by A139, FINSEQ_2:23;
indx (D2,D1,k) in dom D2 by A1, A132, Def19;
then A145: indx (D2,D1,k) in Seg (len (upper_volume (f,D2))) by A140, FINSEQ_1:def_3;
then A146: 1 <= indx (D2,D1,k) by FINSEQ_1:1;
set KD2 = (upper_volume (f,D2)) | (indx (D2,D1,k));
A147: Sum ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = (Sum p2) + (Sum q2) by A144, RVSUM_1:75;
A148: indx (D2,D1,k) <= len (upper_volume (f,D2)) by A145, FINSEQ_1:1;
then A149: len p2 = len ((upper_volume (f,D2)) | (indx (D2,D1,k))) by A142, FINSEQ_1:59;
for i being Nat st 1 <= i & i <= len p2 holds
p2 . i = ((upper_volume (f,D2)) | (indx (D2,D1,k))) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len p2 implies p2 . i = ((upper_volume (f,D2)) | (indx (D2,D1,k))) . i )
assume that
A150: 1 <= i and
A151: i <= len p2 ; ::_thesis: p2 . i = ((upper_volume (f,D2)) | (indx (D2,D1,k))) . i
A152: i in Seg (len p2) by A150, A151, FINSEQ_1:1;
then A153: i in dom ((upper_volume (f,D2)) | (indx (D2,D1,k))) by A149, FINSEQ_1:def_3;
then A154: i in dom ((upper_volume (f,D2)) | (Seg (indx (D2,D1,k)))) by FINSEQ_1:def_15;
A155: dom ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = Seg (len ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_3
.= Seg (indx (D2,D1,(k + 1))) by A141, FINSEQ_1:59 ;
A156: Seg (indx (D2,D1,k)) c= Seg (indx (D2,D1,(k + 1))) by A137, FINSEQ_1:5;
dom ((upper_volume (f,D2)) | (indx (D2,D1,k))) = Seg (len ((upper_volume (f,D2)) | (indx (D2,D1,k)))) by FINSEQ_1:def_3
.= Seg (indx (D2,D1,k)) by A148, FINSEQ_1:59 ;
then i in dom ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A153, A156, A155;
then A157: i in dom ((upper_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_15;
A158: ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) . i = ((upper_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) . i by FINSEQ_1:def_15
.= (upper_volume (f,D2)) . i by A157, FUNCT_1:47 ;
A159: ((upper_volume (f,D2)) | (indx (D2,D1,k))) . i = ((upper_volume (f,D2)) | (Seg (indx (D2,D1,k)))) . i by FINSEQ_1:def_15
.= (upper_volume (f,D2)) . i by A154, FUNCT_1:47 ;
i in dom p2 by A152, FINSEQ_1:def_3;
hence p2 . i = ((upper_volume (f,D2)) | (indx (D2,D1,k))) . i by A144, A159, A158, FINSEQ_1:def_7; ::_thesis: verum
end;
then A160: p2 = (upper_volume (f,D2)) | (indx (D2,D1,k)) by A149, FINSEQ_1:14;
A161: indx (D2,D1,(k + 1)) <= len D2 by A129, FINSEQ_1:1;
A162: ID = (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) by A139;
A163: Sum q1 >= Sum q2
proof
set MD2 = mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))));
set MD1 = mid (D1,(k + 1),(k + 1));
set DD1 = divset (D1,(k + 1));
set g = f | (divset (D1,(k + 1)));
A164: 1 <= (indx (D2,D1,k)) + 1 by NAT_1:11;
reconsider g = f | (divset (D1,(k + 1))) as PartFunc of (divset (D1,(k + 1))),REAL by PARTFUN1:10;
(k + 1) - 1 = k ;
then A165: lower_bound (divset (D1,(k + 1))) = D1 . k by A103, A107, Def4;
D2 . (indx (D2,D1,(k + 1))) = D1 . (k + 1) by A1, A103, Def19;
then A166: D2 . (indx (D2,D1,(k + 1))) = upper_bound (divset (D1,(k + 1))) by A103, A107, Def4;
A167: (indx (D2,D1,k)) + 1 <= indx (D2,D1,(k + 1)) by A134, NAT_1:13;
then A168: (indx (D2,D1,k)) + 1 <= len D2 by A161, XXREAL_0:2;
then (indx (D2,D1,k)) + 1 in Seg (len D2) by A164, FINSEQ_1:1;
then A169: (indx (D2,D1,k)) + 1 in dom D2 by FINSEQ_1:def_3;
then D2 . ((indx (D2,D1,k)) + 1) >= D2 . (indx (D2,D1,k)) by A133, NAT_1:11, SEQ_4:137;
then D2 . ((indx (D2,D1,k)) + 1) >= lower_bound (divset (D1,(k + 1))) by A1, A132, A165, Def19;
then reconsider MD2 = mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1)))) as Division of divset (D1,(k + 1)) by A128, A167, A169, A166, Th37;
A170: ((indx (D2,D1,(k + 1))) -' ((indx (D2,D1,k)) + 1)) + 1 = ((indx (D2,D1,(k + 1))) - ((indx (D2,D1,k)) + 1)) + 1 by A167, XREAL_1:233
.= (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) ;
A171: for n being Nat st 1 <= n & n <= len q2 holds
q2 . n = (upper_volume (g,MD2)) . n
proof
A172: dom ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = Seg (len ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_3
.= Seg (indx (D2,D1,(k + 1))) by A141, FINSEQ_1:59 ;
then A173: dom ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) c= Seg (len D2) by A161, FINSEQ_1:5;
then A174: dom ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) c= dom D2 by FINSEQ_1:def_3;
A175: len (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))))) = ID by A130, A161, A139, A167, A168, A164, A170, FINSEQ_6:118;
let n be Nat; ::_thesis: ( 1 <= n & n <= len q2 implies q2 . n = (upper_volume (g,MD2)) . n )
assume that
A176: 1 <= n and
A177: n <= len q2 ; ::_thesis: q2 . n = (upper_volume (g,MD2)) . n
A178: n in Seg (len q2) by A176, A177, FINSEQ_1:1;
then A179: n in dom q2 by FINSEQ_1:def_3;
then A180: (indx (D2,D1,k)) + n in dom ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A142, A144, FINSEQ_1:28;
then A181: (indx (D2,D1,k)) + n in dom ((upper_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_15;
A182: q2 . n = ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) . ((indx (D2,D1,k)) + n) by A142, A144, A179, FINSEQ_1:def_7
.= ((upper_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) . ((indx (D2,D1,k)) + n) by FINSEQ_1:def_15
.= (upper_volume (f,D2)) . ((indx (D2,D1,k)) + n) by A181, FUNCT_1:47
.= (upper_bound (rng (f | (divset (D2,((indx (D2,D1,k)) + n)))))) * (vol (divset (D2,((indx (D2,D1,k)) + n)))) by A180, A174, Def6 ;
(indx (D2,D1,k)) + n in Seg (len D2) by A180, A173;
then A183: (indx (D2,D1,k)) + n in dom D2 by FINSEQ_1:def_3;
(indx (D2,D1,k)) + n <= indx (D2,D1,(k + 1)) by A172, A180, FINSEQ_1:1;
then A184: n <= ID by A162, XREAL_1:19;
then n in Seg ID by A176, FINSEQ_1:1;
then A185: n in dom MD2 by A175, FINSEQ_1:def_3;
n in Seg (len (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1)))))) by A176, A184, A175, FINSEQ_1:1;
then A186: n in dom (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))))) by FINSEQ_1:def_3;
A187: 1 <= (indx (D2,D1,k)) + n by A172, A180, FINSEQ_1:1;
A188: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
proof
now__::_thesis:_(_divset_(MD2,n)_=_divset_(D2,((indx_(D2,D1,k))_+_n))_&_divset_(MD2,n)_=_divset_(D2,((indx_(D2,D1,k))_+_n))_&_divset_(D2,((indx_(D2,D1,k))_+_n))_c=_divset_(D1,(k_+_1))_)
percases ( n = 1 or n <> 1 ) ;
supposeA189: n = 1 ; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
then A190: (indx (D2,D1,k)) + 1 <= len D2 by A180, A173, FINSEQ_1:1;
A191: 1 <= (indx (D2,D1,k)) + 1 by A172, A180, A189, FINSEQ_1:1;
A192: upper_bound (divset (MD2,1)) = (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))))) . 1 by A185, A189, Def4
.= D2 . (1 + (indx (D2,D1,k))) by A130, A161, A191, A190, FINSEQ_6:118 ;
A193: (indx (D2,D1,k)) + 1 <> 1 by A146, NAT_1:13;
A194: (k + 1) - 1 = k ;
A195: lower_bound (divset (MD2,1)) = lower_bound (divset (D1,(k + 1))) by A185, A189, Def4
.= D1 . k by A103, A107, A194, Def4 ;
A196: divset (D2,((indx (D2,D1,k)) + n)) = [.(lower_bound (divset (D2,((indx (D2,D1,k)) + 1)))),(upper_bound (divset (D2,((indx (D2,D1,k)) + 1)))).] by A189, Th4
.= [.(D2 . (((indx (D2,D1,k)) + 1) - 1)),(upper_bound (divset (D2,((indx (D2,D1,k)) + 1)))).] by A169, A193, Def4
.= [.(D2 . (indx (D2,D1,k))),(D2 . ((indx (D2,D1,k)) + 1)).] by A169, A193, Def4
.= [.(D1 . k),(D2 . ((indx (D2,D1,k)) + 1)).] by A1, A132, Def19 ;
hence divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) by A189, A195, A192, Th4; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
divset (MD2,n) = [.(D1 . k),(D2 . ((indx (D2,D1,k)) + 1)).] by A189, A195, A192, Th4;
hence ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) ) by A185, A196, Th8; ::_thesis: verum
end;
supposeA197: n <> 1 ; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
A198: (indx (D2,D1,k)) + n <> 1
proof
assume (indx (D2,D1,k)) + n = 1 ; ::_thesis: contradiction
then indx (D2,D1,k) = 1 - n ;
then n + 1 <= 1 by A146, XREAL_1:19;
then n <= 1 - 1 by XREAL_1:19;
hence contradiction by A176, NAT_1:3; ::_thesis: verum
end;
A199: divset (D2,((indx (D2,D1,k)) + n)) = [.(lower_bound (divset (D2,((indx (D2,D1,k)) + n)))),(upper_bound (divset (D2,((indx (D2,D1,k)) + n)))).] by Th4
.= [.(D2 . (((indx (D2,D1,k)) + n) - 1)),(upper_bound (divset (D2,((indx (D2,D1,k)) + n)))).] by A183, A198, Def4
.= [.(D2 . (((indx (D2,D1,k)) + n) - 1)),(D2 . ((indx (D2,D1,k)) + n)).] by A183, A198, Def4 ;
n <= n + 1 by NAT_1:12;
then n - 1 <= n by XREAL_1:20;
then A200: n - 1 <= len MD2 by A184, A175, XXREAL_0:2;
consider n1 being Nat such that
A201: n = 1 + n1 by A176, NAT_1:10;
not n is trivial by A176, A197, NAT_2:def_1;
then n >= 1 + 1 by NAT_2:29;
then A202: 1 <= n - 1 by XREAL_1:19;
A203: (indx (D2,D1,k)) + 1 <= indx (D2,D1,(k + 1)) by A134, NAT_1:13;
reconsider n1 = n1 as Element of NAT by ORDINAL1:def_12;
A204: (n1 + ((indx (D2,D1,k)) + 1)) -' 1 = ((indx (D2,D1,k)) + n) - 1 by A187, A201, XREAL_1:233;
A205: (n + ((indx (D2,D1,k)) + 1)) -' 1 = ((n + (indx (D2,D1,k))) + 1) - 1 by NAT_1:11, XREAL_1:233
.= (indx (D2,D1,k)) + n ;
A206: lower_bound (divset (MD2,n)) = MD2 . (n - 1) by A185, A197, Def4
.= D2 . (((indx (D2,D1,k)) + n) - 1) by A130, A161, A168, A164, A203, A201, A204, A202, A200, FINSEQ_6:118 ;
A207: upper_bound (divset (MD2,n)) = MD2 . n by A185, A197, Def4
.= D2 . ((indx (D2,D1,k)) + n) by A130, A161, A168, A164, A176, A178, A184, A175, A203, A205, FINSEQ_6:118 ;
hence divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) by A206, A199, Th4; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
divset (MD2,n) = [.(D2 . (((indx (D2,D1,k)) + n) - 1)),(D2 . ((indx (D2,D1,k)) + n)).] by A206, A207, Th4;
hence ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) ) by A185, A199, Th8; ::_thesis: verum
end;
end;
end;
hence ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) ) ; ::_thesis: verum
end;
then g | (divset (MD2,n)) = f | (divset (D2,((indx (D2,D1,k)) + n))) by FUNCT_1:51;
hence q2 . n = (upper_volume (g,MD2)) . n by A186, A182, A188, Def6; ::_thesis: verum
end;
(k + 1) - 1 = k ;
then A208: lower_bound (divset (D1,(k + 1))) = D1 . k by A103, A107, Def4;
D1 . (k + 1) = upper_bound (divset (D1,(k + 1))) by A103, A107, Def4;
then reconsider MD1 = mid (D1,(k + 1),(k + 1)) as Division of divset (D1,(k + 1)) by A103, A113, A132, A208, Th37, SEQ_4:137;
A209: g | (divset (D1,(k + 1))) is bounded_above
proof
consider a being real number such that
A210: for x being set st x in A /\ (dom f) holds
f . x <= a by A2, RFUNCT_1:70;
for x being set st x in (divset (D1,(k + 1))) /\ (dom g) holds
g . x <= a
proof
let x be set ; ::_thesis: ( x in (divset (D1,(k + 1))) /\ (dom g) implies g . x <= a )
A211: dom g c= dom f by RELAT_1:60;
assume x in (divset (D1,(k + 1))) /\ (dom g) ; ::_thesis: g . x <= a
then A212: x in dom g by XBOOLE_0:def_4;
A213: A /\ (dom f) = dom f by XBOOLE_1:28;
then x in A /\ (dom f) by A212, A211;
then reconsider x = x as Element of A ;
f . x <= a by A210, A212, A213, A211;
hence g . x <= a by A212, FUNCT_1:47; ::_thesis: verum
end;
hence g | (divset (D1,(k + 1))) is bounded_above by RFUNCT_1:70; ::_thesis: verum
end;
len MD1 = ((k + 1) -' (k + 1)) + 1 by A105, A106, FINSEQ_6:118;
then A214: len MD1 = ((k + 1) - (k + 1)) + 1 by XREAL_1:233;
then A215: dom q1 = dom MD1 by A111, FINSEQ_3:29;
A216: for n being Nat st 1 <= n & n <= len q1 holds
q1 . n = (upper_volume (g,MD1)) . n
proof
k + 1 in Seg (len ((upper_volume (f,D1)) | (k + 1))) by A109, FINSEQ_1:4;
then k + 1 in dom ((upper_volume (f,D1)) | (k + 1)) by FINSEQ_1:def_3;
then A217: k + 1 in dom ((upper_volume (f,D1)) | (Seg (k + 1))) by FINSEQ_1:def_15;
A218: MD1 . 1 = D1 . (k + 1) by A105, A106, FINSEQ_6:118;
1 in Seg (len MD1) by A214, FINSEQ_1:3;
then A219: 1 in dom MD1 by FINSEQ_1:def_3;
divset (MD1,1) = [.(lower_bound (divset (MD1,1))),(upper_bound (divset (MD1,1))).] by Th4;
then A220: divset (MD1,1) = [.(lower_bound (divset (D1,(k + 1)))),(upper_bound (divset (MD1,1))).] by A219, Def4
.= [.(lower_bound (divset (D1,(k + 1)))),(D1 . (k + 1)).] by A219, A218, Def4 ;
(k + 1) - 1 = k ;
then A221: lower_bound (divset (D1,(k + 1))) = D1 . k by A103, A107, Def4;
let n be Nat; ::_thesis: ( 1 <= n & n <= len q1 implies q1 . n = (upper_volume (g,MD1)) . n )
assume that
A222: 1 <= n and
A223: n <= len q1 ; ::_thesis: q1 . n = (upper_volume (g,MD1)) . n
A224: n = 1 by A111, A222, A223, XXREAL_0:1;
n in Seg (len q1) by A222, A223, FINSEQ_1:1;
then A225: n in dom q1 by FINSEQ_1:def_3;
upper_bound (divset (D1,(k + 1))) = D1 . (k + 1) by A103, A107, Def4;
then divset (D1,(k + 1)) = [.(D1 . k),(D1 . (k + 1)).] by A221, Th4;
then A226: (upper_volume (g,MD1)) . n = (upper_bound (rng (g | (divset (D1,(k + 1)))))) * (vol (divset (D1,(k + 1)))) by A215, A224, A225, A220, A221, Def6;
((upper_volume (f,D1)) | (k + 1)) . (k + 1) = ((upper_volume (f,D1)) | (Seg (k + 1))) . (k + 1) by FINSEQ_1:def_15
.= (upper_volume (f,D1)) . (k + 1) by A217, FUNCT_1:47 ;
then q1 . n = (upper_volume (f,D1)) . (k + 1) by A110, A112, A224, A225, FINSEQ_1:def_7
.= (upper_bound (rng (f | (divset (D1,(k + 1)))))) * (vol (divset (D1,(k + 1)))) by A103, Def6 ;
hence q1 . n = (upper_volume (g,MD1)) . n by A226; ::_thesis: verum
end;
len q1 = len (upper_volume (g,MD1)) by A111, A214, Def6;
then A227: q1 = upper_volume (g,MD1) by A216, FINSEQ_1:14;
dom g = (dom f) /\ (divset (D1,(k + 1))) by RELAT_1:61;
then dom g = A /\ (divset (D1,(k + 1))) by FUNCT_2:def_1;
then dom g = divset (D1,(k + 1)) by A103, Th8, XBOOLE_1:28;
then A228: g is total by PARTFUN1:def_2;
len MD1 = ((k + 1) -' (k + 1)) + 1 by A105, A106, FINSEQ_6:118;
then len MD1 = ((k + 1) - (k + 1)) + 1 by XREAL_1:233;
then A229: upper_sum (g,MD1) >= upper_sum (g,MD2) by A209, A228, Th30;
len (upper_volume (g,MD2)) = len (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))))) by Def6
.= (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) by A130, A161, A167, A168, A164, A170, FINSEQ_6:118 ;
hence Sum q1 >= Sum q2 by A143, A227, A171, A229, FINSEQ_1:14; ::_thesis: verum
end;
Sum ((upper_volume (f,D1)) | (k + 1)) = (Sum p1) + (Sum q1) by A112, RVSUM_1:75;
then Sum q1 = (Sum ((upper_volume (f,D1)) | (k + 1))) - (Sum p1) ;
then Sum ((upper_volume (f,D1)) | (k + 1)) >= (Sum q2) + (Sum p1) by A163, XREAL_1:19;
then (Sum ((upper_volume (f,D1)) | (k + 1))) - (Sum q2) >= Sum p1 by XREAL_1:19;
then (Sum ((upper_volume (f,D1)) | (k + 1))) - (Sum q2) >= Sum p2 by A102, A131, A127, A160, FINSEQ_1:def_3, XXREAL_0:2;
hence Sum ((upper_volume (f,D1)) | (k + 1)) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A147, XREAL_1:19; ::_thesis: verum
end;
end;
end;
hence Sum ((upper_volume (f,D1)) | (k + 1)) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,(k + 1)))) ; ::_thesis: verum
end;
thus for k being non empty Nat holds S1[k] from NAT_1:sch_10(A3, A101); ::_thesis: verum
end;
hence for i being non empty Element of NAT st i in dom D1 holds
Sum ((upper_volume (f,D1)) | i) >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i))) ; ::_thesis: verum
end;
theorem Th39: :: INTEGRA1:39
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
for i being non empty Element of NAT st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i)))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
for i being non empty Element of NAT st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i)))
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
for i being non empty Element of NAT st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i)))
let f be Function of A,REAL; ::_thesis: ( D1 <= D2 & f | A is bounded_below implies for i being non empty Element of NAT st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i))) )
assume that
A1: D1 <= D2 and
A2: f | A is bounded_below ; ::_thesis: for i being non empty Element of NAT st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i)))
for i being non empty Nat st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i)))
proof
defpred S1[ Nat] means ( $1 in dom D1 implies Sum ((lower_volume (f,D1)) | $1) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,$1))) );
A3: S1[1]
proof
set g = f | (divset (D1,1));
set B = divset (D1,1);
set DD2 = mid (D2,1,(indx (D2,D1,1)));
set DD1 = mid (D1,1,1);
reconsider g = f | (divset (D1,1)) as PartFunc of (divset (D1,1)),REAL by PARTFUN1:10;
A4: dom g = (dom f) /\ (divset (D1,1)) by RELAT_1:61;
assume A5: 1 in dom D1 ; ::_thesis: Sum ((lower_volume (f,D1)) | 1) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,1)))
then A6: D1 . 1 = upper_bound (divset (D1,1)) by Def4;
then A7: D2 . (indx (D2,D1,1)) = upper_bound (divset (D1,1)) by A1, A5, Def19;
lower_bound (divset (D1,1)) <= upper_bound (divset (D1,1)) by SEQ_4:11;
then reconsider DD1 = mid (D1,1,1) as Division of divset (D1,1) by A5, A6, Th37;
1 in Seg (len D1) by A5, FINSEQ_1:def_3;
then A8: 1 <= len D1 by FINSEQ_1:1;
then A9: len (mid (D1,1,1)) = (1 -' 1) + 1 by FINSEQ_6:118;
A10: len (lower_volume (g,DD1)) = len DD1 by Def7
.= 1 by A9, XREAL_1:235 ;
A11: len (mid (D1,1,1)) = 1 by A9, XREAL_1:235;
then A12: len (mid (D1,1,1)) = len (D1 | 1) by A8, FINSEQ_1:59;
for k being Nat st 1 <= k & k <= len (mid (D1,1,1)) holds
(mid (D1,1,1)) . k = (D1 | 1) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (mid (D1,1,1)) implies (mid (D1,1,1)) . k = (D1 | 1) . k )
assume that
A13: 1 <= k and
A14: k <= len (mid (D1,1,1)) ; ::_thesis: (mid (D1,1,1)) . k = (D1 | 1) . k
k in Seg (len (D1 | 1)) by A12, A13, A14, FINSEQ_1:1;
then k in dom (D1 | 1) by FINSEQ_1:def_3;
then k in dom (D1 | (Seg 1)) by FINSEQ_1:def_15;
then A15: (D1 | (Seg 1)) . k = D1 . k by FUNCT_1:47;
A16: k = 1 by A11, A13, A14, XXREAL_0:1;
then (mid (D1,1,1)) . k = D1 . ((1 + 1) - 1) by A8, FINSEQ_6:118;
hence (mid (D1,1,1)) . k = (D1 | 1) . k by A16, A15, FINSEQ_1:def_15; ::_thesis: verum
end;
then A17: mid (D1,1,1) = D1 | 1 by A12, FINSEQ_1:14;
A18: for i being Nat st 1 <= i & i <= len (lower_volume (g,DD1)) holds
(lower_volume (g,DD1)) . i = ((lower_volume (f,D1)) | 1) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len (lower_volume (g,DD1)) implies (lower_volume (g,DD1)) . i = ((lower_volume (f,D1)) | 1) . i )
assume that
A19: 1 <= i and
A20: i <= len (lower_volume (g,DD1)) ; ::_thesis: (lower_volume (g,DD1)) . i = ((lower_volume (f,D1)) | 1) . i
A21: 1 in Seg 1 by FINSEQ_1:3;
dom (D1 | (Seg 1)) = (dom D1) /\ (Seg 1) by RELAT_1:61;
then A22: 1 in dom (D1 | (Seg 1)) by A5, A21, XBOOLE_0:def_4;
dom (lower_volume (f,D1)) = Seg (len (lower_volume (f,D1))) by FINSEQ_1:def_3
.= Seg (len D1) by Def7 ;
then A23: dom ((lower_volume (f,D1)) | (Seg 1)) = (Seg (len D1)) /\ (Seg 1) by RELAT_1:61
.= Seg 1 by A8, FINSEQ_1:7 ;
len DD1 = 1 by A9, XREAL_1:235;
then A24: 1 in dom DD1 by A21, FINSEQ_1:def_3;
A25: ((lower_volume (f,D1)) | 1) . i = ((lower_volume (f,D1)) | (Seg 1)) . i by FINSEQ_1:def_15
.= ((lower_volume (f,D1)) | (Seg 1)) . 1 by A10, A19, A20, XXREAL_0:1
.= (lower_volume (f,D1)) . 1 by A23, FINSEQ_1:3, FUNCT_1:47
.= (lower_bound (rng (f | (divset (D1,1))))) * (vol (divset (D1,1))) by A5, Def7 ;
A26: divset (D1,1) = [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by Th4
.= [.(lower_bound A),(upper_bound (divset (D1,1))).] by A5, Def4
.= [.(lower_bound A),(D1 . 1).] by A5, Def4 ;
A27: (lower_volume (g,DD1)) . i = (lower_volume (g,DD1)) . 1 by A10, A19, A20, XXREAL_0:1
.= (lower_bound (rng (g | (divset (DD1,1))))) * (vol (divset (DD1,1))) by A24, Def7 ;
divset (DD1,1) = [.(lower_bound (divset (DD1,1))),(upper_bound (divset (DD1,1))).] by Th4
.= [.(lower_bound (divset (D1,1))),(upper_bound (divset (DD1,1))).] by A24, Def4
.= [.(lower_bound (divset (D1,1))),(DD1 . 1).] by A24, Def4
.= [.(lower_bound A),((D1 | 1) . 1).] by A5, A17, Def4
.= [.(lower_bound A),((D1 | (Seg 1)) . 1).] by FINSEQ_1:def_15
.= [.(lower_bound A),(D1 . 1).] by A22, FUNCT_1:47 ;
hence (lower_volume (g,DD1)) . i = ((lower_volume (f,D1)) | 1) . i by A4, A27, A26, A25, RELAT_1:68; ::_thesis: verum
end;
A28: g | (divset (D1,1)) is bounded_below
proof
consider a being real number such that
A29: for x being set st x in A /\ (dom f) holds
a <= f . x by A2, RFUNCT_1:71;
for x being set st x in (divset (D1,1)) /\ (dom g) holds
a <= g . x
proof
let x be set ; ::_thesis: ( x in (divset (D1,1)) /\ (dom g) implies a <= g . x )
A30: dom g c= dom f by RELAT_1:60;
assume x in (divset (D1,1)) /\ (dom g) ; ::_thesis: a <= g . x
then A31: x in dom g by XBOOLE_0:def_4;
A32: A /\ (dom f) = dom f by XBOOLE_1:28;
then reconsider x = x as Element of A by A31, A30, XBOOLE_0:def_4;
a <= f . x by A29, A31, A32, A30;
hence a <= g . x by A31, FUNCT_1:47; ::_thesis: verum
end;
hence g | (divset (D1,1)) is bounded_below by RFUNCT_1:71; ::_thesis: verum
end;
A33: rng D2 c= A by Def2;
A34: indx (D2,D1,1) in dom D2 by A1, A5, Def19;
then A35: indx (D2,D1,1) in Seg (len D2) by FINSEQ_1:def_3;
then A36: 1 <= indx (D2,D1,1) by FINSEQ_1:1;
A37: indx (D2,D1,1) <= len D2 by A35, FINSEQ_1:1;
then 1 <= len D2 by A36, XXREAL_0:2;
then 1 in Seg (len D2) by FINSEQ_1:1;
then A38: 1 in dom D2 by FINSEQ_1:def_3;
then D2 . 1 in rng D2 by FUNCT_1:def_3;
then D2 . 1 in A by A33;
then D2 . 1 in [.(lower_bound A),(upper_bound A).] by Th4;
then D2 . 1 in { a where a is Real : ( lower_bound A <= a & a <= upper_bound A ) } by RCOMP_1:def_1;
then ex a being Real st
( D2 . 1 = a & lower_bound A <= a & a <= upper_bound A ) ;
then D2 . 1 >= lower_bound (divset (D1,1)) by A5, Def4;
then reconsider DD2 = mid (D2,1,(indx (D2,D1,1))) as Division of divset (D1,1) by A34, A36, A38, A7, Th37;
indx (D2,D1,1) in dom D2 by A1, A5, Def19;
then A39: indx (D2,D1,1) in Seg (len D2) by FINSEQ_1:def_3;
then A40: 1 <= indx (D2,D1,1) by FINSEQ_1:1;
A41: indx (D2,D1,1) <= len D2 by A39, FINSEQ_1:1;
then A42: 1 <= len D2 by A40, XXREAL_0:2;
then len (mid (D2,1,(indx (D2,D1,1)))) = ((indx (D2,D1,1)) -' 1) + 1 by A40, A41, FINSEQ_6:118;
then A43: len (mid (D2,1,(indx (D2,D1,1)))) = ((indx (D2,D1,1)) - 1) + 1 by A40, XREAL_1:233;
then A44: len (mid (D2,1,(indx (D2,D1,1)))) = len (D2 | (indx (D2,D1,1))) by A41, FINSEQ_1:59;
A45: for k being Nat st 1 <= k & k <= len (mid (D2,1,(indx (D2,D1,1)))) holds
(mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len (mid (D2,1,(indx (D2,D1,1)))) implies (mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k )
assume that
A46: 1 <= k and
A47: k <= len (mid (D2,1,(indx (D2,D1,1)))) ; ::_thesis: (mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k
A48: k in Seg (len (D2 | (indx (D2,D1,1)))) by A44, A46, A47, FINSEQ_1:1;
then k in dom (D2 | (indx (D2,D1,1))) by FINSEQ_1:def_3;
then k in dom (D2 | (Seg (indx (D2,D1,1)))) by FINSEQ_1:def_15;
then A49: (D2 | (Seg (indx (D2,D1,1)))) . k = D2 . k by FUNCT_1:47;
(mid (D2,1,(indx (D2,D1,1)))) . k = D2 . ((k + 1) -' 1) by A40, A41, A42, A46, A47, A48, FINSEQ_6:118;
then (mid (D2,1,(indx (D2,D1,1)))) . k = D2 . ((k + 1) - 1) by NAT_1:11, XREAL_1:233;
hence (mid (D2,1,(indx (D2,D1,1)))) . k = (D2 | (indx (D2,D1,1))) . k by A49, FINSEQ_1:def_15; ::_thesis: verum
end;
then A50: mid (D2,1,(indx (D2,D1,1))) = D2 | (indx (D2,D1,1)) by A44, FINSEQ_1:14;
A51: for i being Nat st 1 <= i & i <= len (lower_volume (g,DD2)) holds
(lower_volume (g,DD2)) . i = ((lower_volume (f,D2)) | (indx (D2,D1,1))) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len (lower_volume (g,DD2)) implies (lower_volume (g,DD2)) . i = ((lower_volume (f,D2)) | (indx (D2,D1,1))) . i )
assume that
A52: 1 <= i and
A53: i <= len (lower_volume (g,DD2)) ; ::_thesis: (lower_volume (g,DD2)) . i = ((lower_volume (f,D2)) | (indx (D2,D1,1))) . i
A54: i <= len DD2 by A53, Def7;
then A55: i in Seg (len DD2) by A52, FINSEQ_1:1;
then A56: i in dom DD2 by FINSEQ_1:def_3;
divset (DD2,i) = divset (D2,i)
proof
Seg (indx (D2,D1,1)) c= Seg (len D2) by A41, FINSEQ_1:5;
then i in Seg (len D2) by A43, A55;
then A57: i in dom D2 by FINSEQ_1:def_3;
now__::_thesis:_divset_(DD2,i)_=_divset_(D2,i)
percases ( i = 1 or i <> 1 ) ;
supposeA58: i = 1 ; ::_thesis: divset (DD2,i) = divset (D2,i)
then A59: 1 in dom (D2 | (Seg (indx (D2,D1,1)))) by A50, A56, FINSEQ_1:def_15;
then 1 in (dom D2) /\ (Seg (indx (D2,D1,1))) by RELAT_1:61;
then A60: 1 in dom D2 by XBOOLE_0:def_4;
A61: divset (D2,i) = [.(lower_bound (divset (D2,1))),(upper_bound (divset (D2,1))).] by A58, Th4
.= [.(lower_bound A),(upper_bound (divset (D2,1))).] by A60, Def4
.= [.(lower_bound A),(D2 . 1).] by A60, Def4 ;
divset (DD2,i) = [.(lower_bound (divset (DD2,1))),(upper_bound (divset (DD2,1))).] by A58, Th4
.= [.(lower_bound (divset (D1,1))),(upper_bound (divset (DD2,1))).] by A56, A58, Def4
.= [.(lower_bound (divset (D1,1))),(DD2 . 1).] by A56, A58, Def4
.= [.(lower_bound (divset (D1,1))),((D2 | (indx (D2,D1,1))) . 1).] by A45, A54, A58
.= [.(lower_bound (divset (D1,1))),((D2 | (Seg (indx (D2,D1,1)))) . 1).] by FINSEQ_1:def_15
.= [.(lower_bound (divset (D1,1))),(D2 . 1).] by A59, FUNCT_1:47
.= [.(lower_bound A),(D2 . 1).] by A5, Def4 ;
hence divset (DD2,i) = divset (D2,i) by A61; ::_thesis: verum
end;
supposeA62: i <> 1 ; ::_thesis: divset (DD2,i) = divset (D2,i)
A63: i - 1 in dom (D2 | (Seg (indx (D2,D1,1))))
proof
not i is trivial by A52, A62, NAT_2:def_1;
then A64: i >= 1 + 1 by NAT_2:29;
then A65: 1 <= i - 1 by XREAL_1:19;
A66: ex j being Nat st i = 1 + j by A52, NAT_1:10;
A67: i - 1 <= (indx (D2,D1,1)) - 0 by A43, A54, XREAL_1:13;
then i - 1 <= len D2 by A37, XXREAL_0:2;
then i - 1 in Seg (len D2) by A66, A65, FINSEQ_1:1;
then A68: i - 1 in dom D2 by FINSEQ_1:def_3;
i - 1 >= 1 by A64, XREAL_1:19;
then i - 1 in Seg (indx (D2,D1,1)) by A66, A67, FINSEQ_1:1;
then i - 1 in (dom D2) /\ (Seg (indx (D2,D1,1))) by A68, XBOOLE_0:def_4;
hence i - 1 in dom (D2 | (Seg (indx (D2,D1,1)))) by RELAT_1:61; ::_thesis: verum
end;
DD2 . (i - 1) = (D2 | (indx (D2,D1,1))) . (i - 1) by A44, A45, FINSEQ_1:14
.= (D2 | (Seg (indx (D2,D1,1)))) . (i - 1) by FINSEQ_1:def_15 ;
then A69: DD2 . (i - 1) = D2 . (i - 1) by A63, FUNCT_1:47;
i <= len D2 by A43, A37, A54, XXREAL_0:2;
then i in Seg (len D2) by A52, FINSEQ_1:1;
then i in dom D2 by FINSEQ_1:def_3;
then i in (dom D2) /\ (Seg (indx (D2,D1,1))) by A43, A55, XBOOLE_0:def_4;
then A70: i in dom (D2 | (Seg (indx (D2,D1,1)))) by RELAT_1:61;
DD2 . i = (D2 | (indx (D2,D1,1))) . i by A44, A45, FINSEQ_1:14
.= (D2 | (Seg (indx (D2,D1,1)))) . i by FINSEQ_1:def_15 ;
then A71: DD2 . i = D2 . i by A70, FUNCT_1:47;
A72: divset (D2,i) = [.(lower_bound (divset (D2,i))),(upper_bound (divset (D2,i))).] by Th4
.= [.(D2 . (i - 1)),(upper_bound (divset (D2,i))).] by A57, A62, Def4
.= [.(D2 . (i - 1)),(D2 . i).] by A57, A62, Def4 ;
divset (DD2,i) = [.(lower_bound (divset (DD2,i))),(upper_bound (divset (DD2,i))).] by Th4
.= [.(DD2 . (i - 1)),(upper_bound (divset (DD2,i))).] by A56, A62, Def4
.= [.(D2 . (i - 1)),(D2 . i).] by A56, A62, A69, A71, Def4 ;
hence divset (DD2,i) = divset (D2,i) by A72; ::_thesis: verum
end;
end;
end;
hence divset (DD2,i) = divset (D2,i) ; ::_thesis: verum
end;
then A73: (lower_volume (g,DD2)) . i = (lower_bound (rng (g | (divset (D2,i))))) * (vol (divset (D2,i))) by A56, Def7;
Seg (indx (D2,D1,1)) c= Seg (len D2) by A41, FINSEQ_1:5;
then i in Seg (len D2) by A43, A55;
then A74: i in dom D2 by FINSEQ_1:def_3;
A75: i in dom DD2 by A55, FINSEQ_1:def_3;
A76: now__::_thesis:_(_lower_bound_(divset_(D2,i))_in_[.(lower_bound_(divset_(D1,1))),(upper_bound_(divset_(D1,1))).]_&_upper_bound_(divset_(D2,i))_in_[.(lower_bound_(divset_(D1,1))),(upper_bound_(divset_(D1,1))).]_)
percases ( i = 1 or i <> 1 ) ;
supposeA77: i = 1 ; ::_thesis: ( lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] & upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] )
then 1 in dom (D2 | (Seg (indx (D2,D1,1)))) by A50, A75, FINSEQ_1:def_15;
then 1 in (dom D2) /\ (Seg (indx (D2,D1,1))) by RELAT_1:61;
then A78: 1 in dom D2 by XBOOLE_0:def_4;
then D2 . 1 <= D2 . (indx (D2,D1,1)) by A34, A36, SEQ_4:137;
then A79: D2 . 1 <= D1 . 1 by A1, A5, Def19;
lower_bound (divset (D2,i)) = lower_bound A by A77, A78, Def4;
then A80: lower_bound (divset (D2,i)) = lower_bound (divset (D1,1)) by A5, Def4;
upper_bound (divset (D2,i)) = D2 . 1 by A77, A78, Def4;
then A81: upper_bound (divset (D2,i)) <= upper_bound (divset (D1,1)) by A5, A79, Def4;
lower_bound (divset (D1,1)) <= upper_bound (divset (D1,1)) by SEQ_4:11;
hence lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by A80, XXREAL_1:1; ::_thesis: upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).]
lower_bound (divset (D2,i)) <= upper_bound (divset (D2,i)) by SEQ_4:11;
then upper_bound (divset (D2,i)) in { r where r is Real : ( lower_bound (divset (D1,1)) <= r & r <= upper_bound (divset (D1,1)) ) } by A80, A81;
hence upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by RCOMP_1:def_1; ::_thesis: verum
end;
supposeA82: i <> 1 ; ::_thesis: ( lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] & upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] )
then not i is trivial by A52, NAT_2:def_1;
then i >= 1 + 1 by NAT_2:29;
then A83: 1 <= i - 1 by XREAL_1:19;
A84: ex j being Nat st i = 1 + j by A52, NAT_1:10;
A85: rng D2 c= A by Def2;
A86: lower_bound (divset (D2,i)) = D2 . (i - 1) by A74, A82, Def4;
A87: lower_bound (divset (D1,1)) = lower_bound A by A5, Def4;
A88: i - 1 <= (indx (D2,D1,1)) - 0 by A43, A54, XREAL_1:13;
then i - 1 <= len D2 by A37, XXREAL_0:2;
then i - 1 in Seg (len D2) by A84, A83, FINSEQ_1:1;
then A89: i - 1 in dom D2 by FINSEQ_1:def_3;
then D2 . (i - 1) in rng D2 by FUNCT_1:def_3;
then A90: lower_bound (divset (D2,i)) >= lower_bound (divset (D1,1)) by A86, A87, A85, SEQ_4:def_2;
A91: upper_bound (divset (D1,1)) = D1 . 1 by A5, Def4;
D2 . (i - 1) <= D2 . (indx (D2,D1,1)) by A34, A88, A89, SEQ_4:137;
then lower_bound (divset (D2,i)) <= upper_bound (divset (D1,1)) by A1, A5, A86, A91, Def19;
then lower_bound (divset (D2,i)) in { r where r is Real : ( lower_bound (divset (D1,1)) <= r & r <= upper_bound (divset (D1,1)) ) } by A90;
hence lower_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by RCOMP_1:def_1; ::_thesis: upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).]
A92: upper_bound (divset (D2,i)) = D2 . i by A74, A82, Def4;
D2 . i in rng D2 by A74, FUNCT_1:def_3;
then A93: upper_bound (divset (D2,i)) >= lower_bound (divset (D1,1)) by A92, A87, A85, SEQ_4:def_2;
D2 . i <= D2 . (indx (D2,D1,1)) by A43, A34, A54, A74, SEQ_4:137;
then upper_bound (divset (D2,i)) <= upper_bound (divset (D1,1)) by A1, A5, A92, A91, Def19;
then upper_bound (divset (D2,i)) in { r where r is Real : ( lower_bound (divset (D1,1)) <= r & r <= upper_bound (divset (D1,1)) ) } by A93;
hence upper_bound (divset (D2,i)) in [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by RCOMP_1:def_1; ::_thesis: verum
end;
end;
end;
A94: divset (D1,1) = [.(lower_bound (divset (D1,1))),(upper_bound (divset (D1,1))).] by Th4;
A95: Seg (indx (D2,D1,1)) c= Seg (len D2) by A41, FINSEQ_1:5;
then i in Seg (len D2) by A43, A55;
then A96: i in dom D2 by FINSEQ_1:def_3;
divset (D2,i) = [.(lower_bound (divset (D2,i))),(upper_bound (divset (D2,i))).] by Th4;
then A97: divset (D2,i) c= divset (D1,1) by A94, A76, XXREAL_2:def_12;
A98: dom ((lower_volume (f,D2)) | (Seg (indx (D2,D1,1)))) = (dom (lower_volume (f,D2))) /\ (Seg (indx (D2,D1,1))) by RELAT_1:61
.= (Seg (len (lower_volume (f,D2)))) /\ (Seg (indx (D2,D1,1))) by FINSEQ_1:def_3
.= (Seg (len D2)) /\ (Seg (indx (D2,D1,1))) by Def7
.= Seg (indx (D2,D1,1)) by A95, XBOOLE_1:28 ;
((lower_volume (f,D2)) | (indx (D2,D1,1))) . i = ((lower_volume (f,D2)) | (Seg (indx (D2,D1,1)))) . i by FINSEQ_1:def_15
.= (lower_volume (f,D2)) . i by A43, A55, A98, FUNCT_1:47
.= (lower_bound (rng (f | (divset (D2,i))))) * (vol (divset (D2,i))) by A96, Def7 ;
hence (lower_volume (g,DD2)) . i = ((lower_volume (f,D2)) | (indx (D2,D1,1))) . i by A73, A97, FUNCT_1:51; ::_thesis: verum
end;
1 <= len (lower_volume (f,D1)) by A8, Def7;
then len (lower_volume (g,DD1)) = len ((lower_volume (f,D1)) | 1) by A10, FINSEQ_1:59;
then A99: lower_volume (g,DD1) = (lower_volume (f,D1)) | 1 by A18, FINSEQ_1:14;
A100: indx (D2,D1,1) <= len (lower_volume (f,D2)) by A41, Def7;
len (lower_volume (g,DD2)) = indx (D2,D1,1) by A43, Def7;
then A101: len (lower_volume (g,DD2)) = len ((lower_volume (f,D2)) | (indx (D2,D1,1))) by A100, FINSEQ_1:59;
dom g = A /\ (divset (D1,1)) by A4, FUNCT_2:def_1;
then dom g = divset (D1,1) by A5, Th8, XBOOLE_1:28;
then g is total by PARTFUN1:def_2;
then lower_sum (g,DD1) <= lower_sum (g,DD2) by A11, A28, Th31;
hence Sum ((lower_volume (f,D1)) | 1) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,1))) by A99, A101, A51, FINSEQ_1:14; ::_thesis: verum
end;
A102: for k being non empty Nat st S1[k] holds
S1[k + 1]
proof
let k be non empty Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A103: ( k in dom D1 implies Sum ((lower_volume (f,D1)) | k) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,k))) ) ; ::_thesis: S1[k + 1]
assume A104: k + 1 in dom D1 ; ::_thesis: Sum ((lower_volume (f,D1)) | (k + 1)) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1))))
then A105: k + 1 in Seg (len D1) by FINSEQ_1:def_3;
then A106: 1 <= k + 1 by FINSEQ_1:1;
A107: k + 1 <= len D1 by A105, FINSEQ_1:1;
now__::_thesis:_Sum_((lower_volume_(f,D1))_|_(k_+_1))_<=_Sum_((lower_volume_(f,D2))_|_(indx_(D2,D1,(k_+_1))))
percases ( 1 = k + 1 or 1 <> k + 1 ) ;
suppose 1 = k + 1 ; ::_thesis: Sum ((lower_volume (f,D1)) | (k + 1)) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1))))
hence Sum ((lower_volume (f,D1)) | (k + 1)) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A3, A104; ::_thesis: verum
end;
supposeA108: 1 <> k + 1 ; ::_thesis: Sum ((lower_volume (f,D1)) | (k + 1)) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1))))
set IDK = indx (D2,D1,k);
set IDK1 = indx (D2,D1,(k + 1));
set K1D2 = (lower_volume (f,D2)) | (indx (D2,D1,(k + 1)));
set KD1 = (lower_volume (f,D1)) | k;
set K1D1 = (lower_volume (f,D1)) | (k + 1);
set n = k + 1;
A109: k + 1 <= len (lower_volume (f,D1)) by A107, Def7;
then A110: len ((lower_volume (f,D1)) | (k + 1)) = k + 1 by FINSEQ_1:59;
then consider p1, q1 being FinSequence of REAL such that
A111: len p1 = k and
A112: len q1 = 1 and
A113: (lower_volume (f,D1)) | (k + 1) = p1 ^ q1 by FINSEQ_2:23;
A114: k <= k + 1 by NAT_1:11;
then A115: k <= len D1 by A107, XXREAL_0:2;
then A116: k <= len (lower_volume (f,D1)) by Def7;
then A117: len p1 = len ((lower_volume (f,D1)) | k) by A111, FINSEQ_1:59;
for i being Nat st 1 <= i & i <= len p1 holds
p1 . i = ((lower_volume (f,D1)) | k) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len p1 implies p1 . i = ((lower_volume (f,D1)) | k) . i )
assume that
A118: 1 <= i and
A119: i <= len p1 ; ::_thesis: p1 . i = ((lower_volume (f,D1)) | k) . i
A120: i in Seg (len p1) by A118, A119, FINSEQ_1:1;
then A121: i in dom ((lower_volume (f,D1)) | k) by A117, FINSEQ_1:def_3;
then A122: i in dom ((lower_volume (f,D1)) | (Seg k)) by FINSEQ_1:def_15;
k <= k + 1 by NAT_1:11;
then A123: Seg k c= Seg (k + 1) by FINSEQ_1:5;
A124: dom ((lower_volume (f,D1)) | (k + 1)) = Seg (len ((lower_volume (f,D1)) | (k + 1))) by FINSEQ_1:def_3
.= Seg (k + 1) by A109, FINSEQ_1:59 ;
dom ((lower_volume (f,D1)) | k) = Seg (len ((lower_volume (f,D1)) | k)) by FINSEQ_1:def_3
.= Seg k by A116, FINSEQ_1:59 ;
then i in dom ((lower_volume (f,D1)) | (k + 1)) by A121, A123, A124;
then A125: i in dom ((lower_volume (f,D1)) | (Seg (k + 1))) by FINSEQ_1:def_15;
A126: ((lower_volume (f,D1)) | (k + 1)) . i = ((lower_volume (f,D1)) | (Seg (k + 1))) . i by FINSEQ_1:def_15
.= (lower_volume (f,D1)) . i by A125, FUNCT_1:47 ;
A127: ((lower_volume (f,D1)) | k) . i = ((lower_volume (f,D1)) | (Seg k)) . i by FINSEQ_1:def_15
.= (lower_volume (f,D1)) . i by A122, FUNCT_1:47 ;
i in dom p1 by A120, FINSEQ_1:def_3;
hence p1 . i = ((lower_volume (f,D1)) | k) . i by A113, A127, A126, FINSEQ_1:def_7; ::_thesis: verum
end;
then A128: p1 = (lower_volume (f,D1)) | k by A117, FINSEQ_1:14;
A129: indx (D2,D1,(k + 1)) in dom D2 by A1, A104, Def19;
then A130: indx (D2,D1,(k + 1)) in Seg (len D2) by FINSEQ_1:def_3;
then A131: 1 <= indx (D2,D1,(k + 1)) by FINSEQ_1:1;
not k + 1 is trivial by A108, NAT_2:def_1;
then k + 1 >= 2 by NAT_2:29;
then k >= (1 + 1) - 1 by XREAL_1:20;
then A132: k in Seg (len D1) by A115, FINSEQ_1:1;
then A133: k in dom D1 by FINSEQ_1:def_3;
then A134: indx (D2,D1,k) in dom D2 by A1, Def19;
A135: indx (D2,D1,k) < indx (D2,D1,(k + 1))
proof
k < k + 1 by NAT_1:13;
then A136: D1 . k < D1 . (k + 1) by A104, A133, SEQM_3:def_1;
assume indx (D2,D1,k) >= indx (D2,D1,(k + 1)) ; ::_thesis: contradiction
then A137: D2 . (indx (D2,D1,k)) >= D2 . (indx (D2,D1,(k + 1))) by A134, A129, SEQ_4:137;
D1 . k = D2 . (indx (D2,D1,k)) by A1, A133, Def19;
hence contradiction by A1, A104, A137, A136, Def19; ::_thesis: verum
end;
A138: indx (D2,D1,(k + 1)) >= indx (D2,D1,k)
proof
assume indx (D2,D1,(k + 1)) < indx (D2,D1,k) ; ::_thesis: contradiction
then A139: D2 . (indx (D2,D1,(k + 1))) < D2 . (indx (D2,D1,k)) by A134, A129, SEQM_3:def_1;
D1 . (k + 1) = D2 . (indx (D2,D1,(k + 1))) by A1, A104, Def19;
then D1 . (k + 1) < D1 . k by A1, A133, A139, Def19;
hence contradiction by A104, A133, NAT_1:11, SEQ_4:137; ::_thesis: verum
end;
then consider ID being Nat such that
A140: indx (D2,D1,(k + 1)) = (indx (D2,D1,k)) + ID by NAT_1:10;
reconsider ID = ID as Element of NAT by ORDINAL1:def_12;
A141: len (lower_volume (f,D2)) = len D2 by Def7;
then A142: indx (D2,D1,(k + 1)) <= len (lower_volume (f,D2)) by A130, FINSEQ_1:1;
then len ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = (indx (D2,D1,k)) + ((indx (D2,D1,(k + 1))) - (indx (D2,D1,k))) by FINSEQ_1:59;
then consider p2, q2 being FinSequence of REAL such that
A143: len p2 = indx (D2,D1,k) and
A144: len q2 = (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) and
A145: (lower_volume (f,D2)) | (indx (D2,D1,(k + 1))) = p2 ^ q2 by A140, FINSEQ_2:23;
A146: indx (D2,D1,(k + 1)) <= len D2 by A130, FINSEQ_1:1;
indx (D2,D1,k) in dom D2 by A1, A133, Def19;
then A147: indx (D2,D1,k) in Seg (len D2) by FINSEQ_1:def_3;
then A148: 1 <= indx (D2,D1,k) by FINSEQ_1:1;
A149: Sum q1 <= Sum q2
proof
set MD2 = mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))));
set MD1 = mid (D1,(k + 1),(k + 1));
set DD1 = divset (D1,(k + 1));
set g = f | (divset (D1,(k + 1)));
A150: 1 <= (indx (D2,D1,k)) + 1 by NAT_1:11;
reconsider g = f | (divset (D1,(k + 1))) as PartFunc of (divset (D1,(k + 1))),REAL by PARTFUN1:10;
(k + 1) - 1 = k ;
then A151: lower_bound (divset (D1,(k + 1))) = D1 . k by A104, A108, Def4;
dom g = (dom f) /\ (divset (D1,(k + 1))) by RELAT_1:61;
then dom g = A /\ (divset (D1,(k + 1))) by FUNCT_2:def_1;
then dom g = divset (D1,(k + 1)) by A104, Th8, XBOOLE_1:28;
then A152: g is total by PARTFUN1:def_2;
A153: upper_bound (divset (D1,(k + 1))) = D1 . (k + 1) by A104, A108, Def4;
A154: D2 . (indx (D2,D1,(k + 1))) = D1 . (k + 1) by A1, A104, Def19;
A155: (indx (D2,D1,k)) + 1 <= indx (D2,D1,(k + 1)) by A135, NAT_1:13;
then A156: (indx (D2,D1,k)) + 1 <= len D2 by A146, XXREAL_0:2;
then (indx (D2,D1,k)) + 1 in Seg (len D2) by A150, FINSEQ_1:1;
then A157: (indx (D2,D1,k)) + 1 in dom D2 by FINSEQ_1:def_3;
then D2 . ((indx (D2,D1,k)) + 1) >= D2 . (indx (D2,D1,k)) by A134, NAT_1:11, SEQ_4:137;
then D2 . ((indx (D2,D1,k)) + 1) >= lower_bound (divset (D1,(k + 1))) by A1, A133, A151, Def19;
then reconsider MD2 = mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1)))) as Division of divset (D1,(k + 1)) by A129, A155, A157, A154, A153, Th37;
A158: ((indx (D2,D1,(k + 1))) -' ((indx (D2,D1,k)) + 1)) + 1 = ((indx (D2,D1,(k + 1))) - ((indx (D2,D1,k)) + 1)) + 1 by A155, XREAL_1:233
.= (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) ;
A159: for n being Nat st 1 <= n & n <= len q2 holds
q2 . n = (lower_volume (g,MD2)) . n
proof
let n be Nat; ::_thesis: ( 1 <= n & n <= len q2 implies q2 . n = (lower_volume (g,MD2)) . n )
assume that
A160: 1 <= n and
A161: n <= len q2 ; ::_thesis: q2 . n = (lower_volume (g,MD2)) . n
A162: n in Seg (len q2) by A160, A161, FINSEQ_1:1;
then A163: n in dom q2 by FINSEQ_1:def_3;
then A164: (indx (D2,D1,k)) + n in dom ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A143, A145, FINSEQ_1:28;
then A165: (indx (D2,D1,k)) + n in dom ((lower_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_15;
A166: len (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))))) = ID by A131, A146, A140, A155, A156, A150, A158, FINSEQ_6:118;
A167: dom ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = Seg (len ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_3
.= Seg (indx (D2,D1,(k + 1))) by A142, FINSEQ_1:59 ;
then (indx (D2,D1,k)) + n <= indx (D2,D1,(k + 1)) by A164, FINSEQ_1:1;
then A168: n <= (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) by XREAL_1:19;
then n in Seg (len (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1)))))) by A140, A160, A166, FINSEQ_1:1;
then A169: n in dom MD2 by FINSEQ_1:def_3;
A170: Seg (indx (D2,D1,(k + 1))) c= Seg (len D2) by A146, FINSEQ_1:5;
then (indx (D2,D1,k)) + n in Seg (len D2) by A167, A164;
then A171: (indx (D2,D1,k)) + n in dom D2 by FINSEQ_1:def_3;
A172: q2 . n = ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) . ((indx (D2,D1,k)) + n) by A143, A145, A163, FINSEQ_1:def_7
.= ((lower_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) . ((indx (D2,D1,k)) + n) by FINSEQ_1:def_15
.= (lower_volume (f,D2)) . ((indx (D2,D1,k)) + n) by A165, FUNCT_1:47
.= (lower_bound (rng (f | (divset (D2,((indx (D2,D1,k)) + n)))))) * (vol (divset (D2,((indx (D2,D1,k)) + n)))) by A171, Def7 ;
A173: 1 <= (indx (D2,D1,k)) + n by A167, A164, FINSEQ_1:1;
A174: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
proof
now__::_thesis:_(_divset_(MD2,n)_=_divset_(D2,((indx_(D2,D1,k))_+_n))_&_divset_(MD2,n)_=_divset_(D2,((indx_(D2,D1,k))_+_n))_&_divset_(D2,((indx_(D2,D1,k))_+_n))_c=_divset_(D1,(k_+_1))_)
percases ( n = 1 or n <> 1 ) ;
supposeA175: n = 1 ; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
then A176: 1 <= (indx (D2,D1,k)) + 1 by A167, A164, FINSEQ_1:1;
A177: (indx (D2,D1,k)) + 1 <= len D2 by A167, A164, A170, A175, FINSEQ_1:1;
A178: upper_bound (divset (MD2,1)) = (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))))) . 1 by A169, A175, Def4
.= D2 . (1 + (indx (D2,D1,k))) by A131, A146, A176, A177, FINSEQ_6:118 ;
A179: (indx (D2,D1,k)) + 1 <> 1 by A148, NAT_1:13;
A180: (k + 1) - 1 = k ;
A181: lower_bound (divset (MD2,1)) = lower_bound (divset (D1,(k + 1))) by A169, A175, Def4
.= D1 . k by A104, A108, A180, Def4 ;
A182: divset (D2,((indx (D2,D1,k)) + n)) = [.(lower_bound (divset (D2,((indx (D2,D1,k)) + 1)))),(upper_bound (divset (D2,((indx (D2,D1,k)) + 1)))).] by A175, Th4
.= [.(D2 . (((indx (D2,D1,k)) + 1) - 1)),(upper_bound (divset (D2,((indx (D2,D1,k)) + 1)))).] by A157, A179, Def4
.= [.(D2 . (indx (D2,D1,k))),(D2 . ((indx (D2,D1,k)) + 1)).] by A157, A179, Def4
.= [.(D1 . k),(D2 . ((indx (D2,D1,k)) + 1)).] by A1, A133, Def19 ;
hence divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) by A175, A181, A178, Th4; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
divset (MD2,n) = [.(D1 . k),(D2 . ((indx (D2,D1,k)) + 1)).] by A175, A181, A178, Th4;
hence ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) ) by A169, A182, Th8; ::_thesis: verum
end;
supposeA183: n <> 1 ; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
A184: (indx (D2,D1,k)) + n <> 1
proof
assume (indx (D2,D1,k)) + n = 1 ; ::_thesis: contradiction
then indx (D2,D1,k) = 1 - n ;
then n + 1 <= 1 by A148, XREAL_1:19;
then n <= 1 - 1 by XREAL_1:19;
hence contradiction by A160, NAT_1:3; ::_thesis: verum
end;
A185: divset (D2,((indx (D2,D1,k)) + n)) = [.(lower_bound (divset (D2,((indx (D2,D1,k)) + n)))),(upper_bound (divset (D2,((indx (D2,D1,k)) + n)))).] by Th4
.= [.(D2 . (((indx (D2,D1,k)) + n) - 1)),(upper_bound (divset (D2,((indx (D2,D1,k)) + n)))).] by A171, A184, Def4
.= [.(D2 . (((indx (D2,D1,k)) + n) - 1)),(D2 . ((indx (D2,D1,k)) + n)).] by A171, A184, Def4 ;
n <= n + 1 by NAT_1:12;
then n - 1 <= n by XREAL_1:20;
then A186: n - 1 <= len MD2 by A140, A168, A166, XXREAL_0:2;
A187: (indx (D2,D1,k)) + 1 <= indx (D2,D1,(k + 1)) by A135, NAT_1:13;
not n is trivial by A160, A183, NAT_2:def_1;
then n >= 1 + 1 by NAT_2:29;
then A188: 1 <= n - 1 by XREAL_1:19;
consider n1 being Nat such that
A189: n = 1 + n1 by A160, NAT_1:10;
reconsider n1 = n1 as Element of NAT by ORDINAL1:def_12;
A190: (n1 + ((indx (D2,D1,k)) + 1)) -' 1 = ((indx (D2,D1,k)) + n) - 1 by A173, A189, XREAL_1:233;
A191: (n + ((indx (D2,D1,k)) + 1)) -' 1 = ((n + (indx (D2,D1,k))) + 1) - 1 by NAT_1:11, XREAL_1:233
.= (indx (D2,D1,k)) + n ;
A192: lower_bound (divset (MD2,n)) = MD2 . (n - 1) by A169, A183, Def4
.= D2 . (((indx (D2,D1,k)) + n) - 1) by A131, A146, A156, A150, A187, A189, A190, A188, A186, FINSEQ_6:118 ;
A193: upper_bound (divset (MD2,n)) = MD2 . n by A169, A183, Def4
.= D2 . ((indx (D2,D1,k)) + n) by A131, A146, A140, A156, A150, A160, A162, A168, A166, A187, A191, FINSEQ_6:118 ;
hence divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) by A192, A185, Th4; ::_thesis: ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) )
divset (MD2,n) = [.(D2 . (((indx (D2,D1,k)) + n) - 1)),(D2 . ((indx (D2,D1,k)) + n)).] by A192, A193, Th4;
hence ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) ) by A169, A185, Th8; ::_thesis: verum
end;
end;
end;
hence ( divset (MD2,n) = divset (D2,((indx (D2,D1,k)) + n)) & divset (D2,((indx (D2,D1,k)) + n)) c= divset (D1,(k + 1)) ) ; ::_thesis: verum
end;
then g | (divset (MD2,n)) = f | (divset (D2,((indx (D2,D1,k)) + n))) by FUNCT_1:51;
hence q2 . n = (lower_volume (g,MD2)) . n by A169, A172, A174, Def7; ::_thesis: verum
end;
(k + 1) - 1 = k ;
then A194: lower_bound (divset (D1,(k + 1))) = D1 . k by A104, A108, Def4;
D1 . (k + 1) = upper_bound (divset (D1,(k + 1))) by A104, A108, Def4;
then reconsider MD1 = mid (D1,(k + 1),(k + 1)) as Division of divset (D1,(k + 1)) by A104, A114, A133, A194, Th37, SEQ_4:137;
A195: g | (divset (D1,(k + 1))) is bounded_below
proof
consider a being real number such that
A196: for x being set st x in A /\ (dom f) holds
a <= f . x by A2, RFUNCT_1:71;
for x being set st x in (divset (D1,(k + 1))) /\ (dom g) holds
a <= g . x
proof
let x be set ; ::_thesis: ( x in (divset (D1,(k + 1))) /\ (dom g) implies a <= g . x )
A197: dom g c= dom f by RELAT_1:60;
assume x in (divset (D1,(k + 1))) /\ (dom g) ; ::_thesis: a <= g . x
then A198: x in dom g by XBOOLE_0:def_4;
A199: A /\ (dom f) = dom f by XBOOLE_1:28;
then reconsider x = x as Element of A by A198, A197, XBOOLE_0:def_4;
a <= f . x by A196, A198, A199, A197;
hence a <= g . x by A198, FUNCT_1:47; ::_thesis: verum
end;
hence g | (divset (D1,(k + 1))) is bounded_below by RFUNCT_1:71; ::_thesis: verum
end;
len MD1 = ((k + 1) -' (k + 1)) + 1 by A106, A107, FINSEQ_6:118;
then A200: len MD1 = ((k + 1) - (k + 1)) + 1 by XREAL_1:233;
A201: for n being Nat st 1 <= n & n <= len q1 holds
q1 . n = (lower_volume (g,MD1)) . n
proof
k + 1 in Seg (len ((lower_volume (f,D1)) | (k + 1))) by A110, FINSEQ_1:4;
then k + 1 in dom ((lower_volume (f,D1)) | (k + 1)) by FINSEQ_1:def_3;
then A202: k + 1 in dom ((lower_volume (f,D1)) | (Seg (k + 1))) by FINSEQ_1:def_15;
A203: ((lower_volume (f,D1)) | (k + 1)) . (k + 1) = ((lower_volume (f,D1)) | (Seg (k + 1))) . (k + 1) by FINSEQ_1:def_15
.= (lower_volume (f,D1)) . (k + 1) by A202, FUNCT_1:47 ;
A204: MD1 . 1 = D1 . (k + 1) by A106, A107, FINSEQ_6:118;
1 in Seg 1 by FINSEQ_1:3;
then A205: 1 in dom MD1 by A200, FINSEQ_1:def_3;
then A206: upper_bound (divset (MD1,1)) = MD1 . 1 by Def4;
let n be Nat; ::_thesis: ( 1 <= n & n <= len q1 implies q1 . n = (lower_volume (g,MD1)) . n )
assume that
A207: 1 <= n and
A208: n <= len q1 ; ::_thesis: q1 . n = (lower_volume (g,MD1)) . n
A209: n = 1 by A112, A207, A208, XXREAL_0:1;
lower_bound (divset (MD1,1)) = lower_bound (divset (D1,(k + 1))) by A205, Def4;
then A210: divset (MD1,1) = [.(lower_bound (divset (D1,(k + 1)))),(D1 . (k + 1)).] by A206, A204, Th4;
(k + 1) - 1 = k ;
then A211: lower_bound (divset (D1,(k + 1))) = D1 . k by A104, A108, Def4;
upper_bound (divset (D1,(k + 1))) = D1 . (k + 1) by A104, A108, Def4;
then A212: divset (D1,(k + 1)) = [.(D1 . k),(D1 . (k + 1)).] by A211, Th4;
A213: n in Seg (len q1) by A207, A208, FINSEQ_1:1;
then n in dom MD1 by A112, A200, FINSEQ_1:def_3;
then A214: (lower_volume (g,MD1)) . n = (lower_bound (rng (g | (divset (D1,(k + 1)))))) * (vol (divset (D1,(k + 1)))) by A209, A210, A211, A212, Def7;
n in dom q1 by A213, FINSEQ_1:def_3;
then q1 . n = (lower_volume (f,D1)) . (k + 1) by A111, A113, A209, A203, FINSEQ_1:def_7
.= (lower_bound (rng (f | (divset (D1,(k + 1)))))) * (vol (divset (D1,(k + 1)))) by A104, Def7 ;
hence q1 . n = (lower_volume (g,MD1)) . n by A214; ::_thesis: verum
end;
len q1 = len (lower_volume (g,MD1)) by A112, A200, Def7;
then A215: q1 = lower_volume (g,MD1) by A201, FINSEQ_1:14;
len MD1 = ((k + 1) -' (k + 1)) + 1 by A106, A107, FINSEQ_6:118;
then len MD1 = ((k + 1) - (k + 1)) + 1 by XREAL_1:233;
then A216: lower_sum (g,MD1) <= lower_sum (g,MD2) by A195, A152, Th31;
len (lower_volume (g,MD2)) = len (mid (D2,((indx (D2,D1,k)) + 1),(indx (D2,D1,(k + 1))))) by Def7
.= (indx (D2,D1,(k + 1))) - (indx (D2,D1,k)) by A131, A146, A155, A156, A150, A158, FINSEQ_6:118 ;
hence Sum q1 <= Sum q2 by A144, A215, A159, A216, FINSEQ_1:14; ::_thesis: verum
end;
set KD2 = (lower_volume (f,D2)) | (indx (D2,D1,k));
A217: Sum ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = (Sum p2) + (Sum q2) by A145, RVSUM_1:75;
A218: indx (D2,D1,k) <= len (lower_volume (f,D2)) by A141, A147, FINSEQ_1:1;
then A219: len p2 = len ((lower_volume (f,D2)) | (indx (D2,D1,k))) by A143, FINSEQ_1:59;
for i being Nat st 1 <= i & i <= len p2 holds
p2 . i = ((lower_volume (f,D2)) | (indx (D2,D1,k))) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len p2 implies p2 . i = ((lower_volume (f,D2)) | (indx (D2,D1,k))) . i )
assume that
A220: 1 <= i and
A221: i <= len p2 ; ::_thesis: p2 . i = ((lower_volume (f,D2)) | (indx (D2,D1,k))) . i
A222: i in Seg (len p2) by A220, A221, FINSEQ_1:1;
then A223: i in dom ((lower_volume (f,D2)) | (indx (D2,D1,k))) by A219, FINSEQ_1:def_3;
then A224: i in dom ((lower_volume (f,D2)) | (Seg (indx (D2,D1,k)))) by FINSEQ_1:def_15;
A225: dom ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) = Seg (len ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_3
.= Seg (indx (D2,D1,(k + 1))) by A142, FINSEQ_1:59 ;
A226: Seg (indx (D2,D1,k)) c= Seg (indx (D2,D1,(k + 1))) by A138, FINSEQ_1:5;
dom ((lower_volume (f,D2)) | (indx (D2,D1,k))) = Seg (len ((lower_volume (f,D2)) | (indx (D2,D1,k)))) by FINSEQ_1:def_3
.= Seg (indx (D2,D1,k)) by A218, FINSEQ_1:59 ;
then i in dom ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A223, A226, A225;
then A227: i in dom ((lower_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) by FINSEQ_1:def_15;
A228: ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) . i = ((lower_volume (f,D2)) | (Seg (indx (D2,D1,(k + 1))))) . i by FINSEQ_1:def_15
.= (lower_volume (f,D2)) . i by A227, FUNCT_1:47 ;
A229: ((lower_volume (f,D2)) | (indx (D2,D1,k))) . i = ((lower_volume (f,D2)) | (Seg (indx (D2,D1,k)))) . i by FINSEQ_1:def_15
.= (lower_volume (f,D2)) . i by A224, FUNCT_1:47 ;
i in dom p2 by A222, FINSEQ_1:def_3;
hence p2 . i = ((lower_volume (f,D2)) | (indx (D2,D1,k))) . i by A145, A229, A228, FINSEQ_1:def_7; ::_thesis: verum
end;
then A230: p2 = (lower_volume (f,D2)) | (indx (D2,D1,k)) by A219, FINSEQ_1:14;
Sum ((lower_volume (f,D1)) | (k + 1)) = (Sum p1) + (Sum q1) by A113, RVSUM_1:75;
then Sum q1 = (Sum ((lower_volume (f,D1)) | (k + 1))) - (Sum p1) ;
then Sum ((lower_volume (f,D1)) | (k + 1)) <= (Sum q2) + (Sum p1) by A149, XREAL_1:20;
then (Sum ((lower_volume (f,D1)) | (k + 1))) - (Sum q2) <= Sum p1 by XREAL_1:20;
then (Sum ((lower_volume (f,D1)) | (k + 1))) - (Sum q2) <= Sum p2 by A103, A132, A128, A230, FINSEQ_1:def_3, XXREAL_0:2;
hence Sum ((lower_volume (f,D1)) | (k + 1)) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) by A217, XREAL_1:20; ::_thesis: verum
end;
end;
end;
hence Sum ((lower_volume (f,D1)) | (k + 1)) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,(k + 1)))) ; ::_thesis: verum
end;
thus for n being non empty Nat holds S1[n] from NAT_1:sch_10(A3, A102); ::_thesis: verum
end;
hence for i being non empty Element of NAT st i in dom D1 holds
Sum ((lower_volume (f,D1)) | i) <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i))) ; ::_thesis: verum
end;
theorem Th40: :: INTEGRA1:40
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_above holds
(PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i))
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_above holds
(PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i))
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_above holds
(PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i))
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_above holds
(PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i))
let f be Function of A,REAL; ::_thesis: ( D1 <= D2 & i in dom D1 & f | A is bounded_above implies (PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i)) )
assume that
A1: D1 <= D2 and
A2: i in dom D1 and
A3: f | A is bounded_above ; ::_thesis: (PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i))
A4: len (upper_volume (f,D2)) = len D2 by Def6;
i in Seg (len D1) by A2, FINSEQ_1:def_3;
then i in Seg (len (upper_volume (f,D1))) by Def6;
then i in dom (upper_volume (f,D1)) by FINSEQ_1:def_3;
then A5: (PartSums (upper_volume (f,D1))) . i = Sum ((upper_volume (f,D1)) | i) by Def20;
indx (D2,D1,i) in dom D2 by A1, A2, Def19;
then indx (D2,D1,i) in Seg (len (upper_volume (f,D2))) by A4, FINSEQ_1:def_3;
then A6: indx (D2,D1,i) in dom (upper_volume (f,D2)) by FINSEQ_1:def_3;
i in Seg (len D1) by A2, FINSEQ_1:def_3;
then i is non empty Element of NAT by FINSEQ_1:1;
then (PartSums (upper_volume (f,D1))) . i >= Sum ((upper_volume (f,D2)) | (indx (D2,D1,i))) by A1, A2, A3, A5, Th38;
hence (PartSums (upper_volume (f,D1))) . i >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,i)) by A6, Def20; ::_thesis: verum
end;
theorem Th41: :: INTEGRA1:41
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i))
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i))
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i))
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st D1 <= D2 & i in dom D1 & f | A is bounded_below holds
(PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i))
let f be Function of A,REAL; ::_thesis: ( D1 <= D2 & i in dom D1 & f | A is bounded_below implies (PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i)) )
assume that
A1: D1 <= D2 and
A2: i in dom D1 and
A3: f | A is bounded_below ; ::_thesis: (PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i))
A4: len (lower_volume (f,D2)) = len D2 by Def7;
i in Seg (len D1) by A2, FINSEQ_1:def_3;
then i in Seg (len (lower_volume (f,D1))) by Def7;
then i in dom (lower_volume (f,D1)) by FINSEQ_1:def_3;
then A5: (PartSums (lower_volume (f,D1))) . i = Sum ((lower_volume (f,D1)) | i) by Def20;
indx (D2,D1,i) in dom D2 by A1, A2, Def19;
then indx (D2,D1,i) in Seg (len (lower_volume (f,D2))) by A4, FINSEQ_1:def_3;
then A6: indx (D2,D1,i) in dom (lower_volume (f,D2)) by FINSEQ_1:def_3;
i in Seg (len D1) by A2, FINSEQ_1:def_3;
then i is non empty Element of NAT by FINSEQ_1:1;
then (PartSums (lower_volume (f,D1))) . i <= Sum ((lower_volume (f,D2)) | (indx (D2,D1,i))) by A1, A2, A3, A5, Th39;
hence (PartSums (lower_volume (f,D1))) . i <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,i)) by A6, Def20; ::_thesis: verum
end;
theorem Th42: :: INTEGRA1:42
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL holds (PartSums (upper_volume (f,D))) . (len D) = upper_sum (f,D)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL holds (PartSums (upper_volume (f,D))) . (len D) = upper_sum (f,D)
let D be Division of A; ::_thesis: for f being Function of A,REAL holds (PartSums (upper_volume (f,D))) . (len D) = upper_sum (f,D)
let f be Function of A,REAL; ::_thesis: (PartSums (upper_volume (f,D))) . (len D) = upper_sum (f,D)
len (upper_volume (f,D)) = len D by Def6;
then len D in Seg (len (upper_volume (f,D))) by FINSEQ_1:3;
then len D in dom (upper_volume (f,D)) by FINSEQ_1:def_3;
then A1: (PartSums (upper_volume (f,D))) . (len D) = Sum ((upper_volume (f,D)) | (len D)) by Def20;
dom (upper_volume (f,D)) = Seg (len (upper_volume (f,D))) by FINSEQ_1:def_3;
then dom (upper_volume (f,D)) = Seg (len D) by Def6;
then (upper_volume (f,D)) | (Seg (len D)) = upper_volume (f,D) by RELAT_1:68;
hence (PartSums (upper_volume (f,D))) . (len D) = upper_sum (f,D) by A1, FINSEQ_1:def_15; ::_thesis: verum
end;
theorem Th43: :: INTEGRA1:43
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being Function of A,REAL holds (PartSums (lower_volume (f,D))) . (len D) = lower_sum (f,D)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f being Function of A,REAL holds (PartSums (lower_volume (f,D))) . (len D) = lower_sum (f,D)
let D be Division of A; ::_thesis: for f being Function of A,REAL holds (PartSums (lower_volume (f,D))) . (len D) = lower_sum (f,D)
let f be Function of A,REAL; ::_thesis: (PartSums (lower_volume (f,D))) . (len D) = lower_sum (f,D)
len (lower_volume (f,D)) = len D by Def7;
then len D in Seg (len (lower_volume (f,D))) by FINSEQ_1:3;
then len D in dom (lower_volume (f,D)) by FINSEQ_1:def_3;
then A1: (PartSums (lower_volume (f,D))) . (len D) = Sum ((lower_volume (f,D)) | (len D)) by Def20;
dom (lower_volume (f,D)) = Seg (len (lower_volume (f,D))) by FINSEQ_1:def_3;
then dom (lower_volume (f,D)) = Seg (len D) by Def7;
then (lower_volume (f,D)) | (Seg (len D)) = lower_volume (f,D) by RELAT_1:68;
hence (PartSums (lower_volume (f,D))) . (len D) = lower_sum (f,D) by A1, FINSEQ_1:def_15; ::_thesis: verum
end;
theorem Th44: :: INTEGRA1:44
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A st D1 <= D2 holds
indx (D2,D1,(len D1)) = len D2
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A st D1 <= D2 holds
indx (D2,D1,(len D1)) = len D2
let D1, D2 be Division of A; ::_thesis: ( D1 <= D2 implies indx (D2,D1,(len D1)) = len D2 )
len D1 in Seg (len D1) by FINSEQ_1:3;
then A1: len D1 in dom D1 by FINSEQ_1:def_3;
assume A2: D1 <= D2 ; ::_thesis: indx (D2,D1,(len D1)) = len D2
then D1 . (len D1) = D2 . (indx (D2,D1,(len D1))) by A1, Def19;
then A3: D2 . (indx (D2,D1,(len D1))) = upper_bound A by Def2;
len D2 in Seg (len D2) by FINSEQ_1:3;
then A4: len D2 in dom D2 by FINSEQ_1:def_3;
assume A5: indx (D2,D1,(len D1)) <> len D2 ; ::_thesis: contradiction
A6: indx (D2,D1,(len D1)) in dom D2 by A2, A1, Def19;
then indx (D2,D1,(len D1)) in Seg (len D2) by FINSEQ_1:def_3;
then indx (D2,D1,(len D1)) <= len D2 by FINSEQ_1:1;
then indx (D2,D1,(len D1)) < len D2 by A5, XXREAL_0:1;
then D2 . (indx (D2,D1,(len D1))) < D2 . (len D2) by A4, A6, SEQM_3:def_1;
hence contradiction by A3, Def2; ::_thesis: verum
end;
theorem Th45: :: INTEGRA1:45
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
upper_sum (f,D2) <= upper_sum (f,D1)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
upper_sum (f,D2) <= upper_sum (f,D1)
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st D1 <= D2 & f | A is bounded_above holds
upper_sum (f,D2) <= upper_sum (f,D1)
let f be Function of A,REAL; ::_thesis: ( D1 <= D2 & f | A is bounded_above implies upper_sum (f,D2) <= upper_sum (f,D1) )
assume that
A1: D1 <= D2 and
A2: f | A is bounded_above ; ::_thesis: upper_sum (f,D2) <= upper_sum (f,D1)
len D1 in Seg (len D1) by FINSEQ_1:3;
then len D1 in dom D1 by FINSEQ_1:def_3;
then (PartSums (upper_volume (f,D1))) . (len D1) >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,(len D1))) by A1, A2, Th40;
then upper_sum (f,D1) >= (PartSums (upper_volume (f,D2))) . (indx (D2,D1,(len D1))) by Th42;
then upper_sum (f,D1) >= (PartSums (upper_volume (f,D2))) . (len D2) by A1, Th44;
hence upper_sum (f,D2) <= upper_sum (f,D1) by Th42; ::_thesis: verum
end;
theorem Th46: :: INTEGRA1:46
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
lower_sum (f,D2) >= lower_sum (f,D1)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
lower_sum (f,D2) >= lower_sum (f,D1)
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st D1 <= D2 & f | A is bounded_below holds
lower_sum (f,D2) >= lower_sum (f,D1)
let f be Function of A,REAL; ::_thesis: ( D1 <= D2 & f | A is bounded_below implies lower_sum (f,D2) >= lower_sum (f,D1) )
assume that
A1: D1 <= D2 and
A2: f | A is bounded_below ; ::_thesis: lower_sum (f,D2) >= lower_sum (f,D1)
len D1 in Seg (len D1) by FINSEQ_1:3;
then len D1 in dom D1 by FINSEQ_1:def_3;
then (PartSums (lower_volume (f,D1))) . (len D1) <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,(len D1))) by A1, A2, Th41;
then lower_sum (f,D1) <= (PartSums (lower_volume (f,D2))) . (indx (D2,D1,(len D1))) by Th43;
then lower_sum (f,D1) <= (PartSums (lower_volume (f,D2))) . (len D2) by A1, Th44;
hence lower_sum (f,D2) >= lower_sum (f,D1) by Th43; ::_thesis: verum
end;
theorem Th47: :: INTEGRA1:47
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A ex D being Division of A st
( D1 <= D & D2 <= D )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A ex D being Division of A st
( D1 <= D & D2 <= D )
let D1, D2 be Division of A; ::_thesis: ex D being Division of A st
( D1 <= D & D2 <= D )
consider D3 being FinSequence of REAL such that
A1: rng D3 = rng (D1 ^ D2) and
A2: len D3 = card (rng (D1 ^ D2)) and
A3: D3 is increasing by SEQ_4:140;
reconsider D3 = D3 as non empty increasing FinSequence of REAL by A1, A3;
A4: rng D2 c= A by Def2;
rng D1 c= A by Def2;
then (rng D1) \/ (rng D2) c= A by A4, XBOOLE_1:8;
then A5: rng D3 c= A by A1, FINSEQ_1:31;
D3 . (len D3) = upper_bound A
proof
assume A6: D3 . (len D3) <> upper_bound A ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( D3 . (len D3) > upper_bound A or D3 . (len D3) < upper_bound A ) by A6, XXREAL_0:1;
supposeA7: D3 . (len D3) > upper_bound A ; ::_thesis: contradiction
len D3 in Seg (len D3) by FINSEQ_1:3;
then len D3 in dom D3 by FINSEQ_1:def_3;
then D3 . (len D3) in rng D3 by FUNCT_1:def_3;
then D3 . (len D3) in A by A5;
then D3 . (len D3) in [.(lower_bound A),(upper_bound A).] by Th4;
then D3 . (len D3) in { r where r is Real : ( lower_bound A <= r & r <= upper_bound A ) } by RCOMP_1:def_1;
then ex a being Real st
( a = D3 . (len D3) & lower_bound A <= a & a <= upper_bound A ) ;
hence contradiction by A7; ::_thesis: verum
end;
supposeA8: D3 . (len D3) < upper_bound A ; ::_thesis: contradiction
A9: rng D1 c= rng (D1 ^ D2) by FINSEQ_1:29;
len D1 in Seg (len D1) by FINSEQ_1:3;
then A10: len D1 in dom D1 by FINSEQ_1:def_3;
len D3 in Seg (len D3) by FINSEQ_1:3;
then A11: len D3 in dom D3 by FINSEQ_1:def_3;
D1 . (len D1) = upper_bound A by Def2;
then upper_bound A in rng D1 by A10, FUNCT_1:def_3;
then consider k being Nat such that
A12: k in dom D3 and
A13: D3 . k = upper_bound A by A1, A9, FINSEQ_2:10;
k in Seg (len D3) by A12, FINSEQ_1:def_3;
then k <= len D3 by FINSEQ_1:1;
hence contradiction by A8, A11, A12, A13, SEQ_4:137; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
then reconsider D3 = D3 as Division of A by A5, Def2;
len D2 = card (rng D2) by FINSEQ_4:62;
then A14: len D2 <= len D3 by A2, FINSEQ_1:30, NAT_1:43;
take D3 ; ::_thesis: ( D1 <= D3 & D2 <= D3 )
A15: rng D1 c= rng (D1 ^ D2) by FINSEQ_1:29;
A16: rng D2 c= rng (D1 ^ D2) by FINSEQ_1:30;
len D1 = card (rng D1) by FINSEQ_4:62;
then len D1 <= len D3 by A2, FINSEQ_1:29, NAT_1:43;
hence ( D1 <= D3 & D2 <= D3 ) by A1, A15, A16, A14, Def18; ::_thesis: verum
end;
theorem Th48: :: INTEGRA1:48
for A being non empty closed_interval Subset of REAL
for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D1) <= upper_sum (f,D2)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D1, D2 being Division of A
for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D1) <= upper_sum (f,D2)
let D1, D2 be Division of A; ::_thesis: for f being Function of A,REAL st f | A is bounded holds
lower_sum (f,D1) <= upper_sum (f,D2)
let f be Function of A,REAL; ::_thesis: ( f | A is bounded implies lower_sum (f,D1) <= upper_sum (f,D2) )
consider D being Division of A such that
A1: D1 <= D and
A2: D2 <= D by Th47;
assume A3: f | A is bounded ; ::_thesis: lower_sum (f,D1) <= upper_sum (f,D2)
then A4: lower_sum (f,D) <= upper_sum (f,D) by Th28;
upper_sum (f,D) <= upper_sum (f,D2) by A3, A2, Th45;
then A5: lower_sum (f,D) <= upper_sum (f,D2) by A4, XXREAL_0:2;
lower_sum (f,D1) <= lower_sum (f,D) by A3, A1, Th46;
hence lower_sum (f,D1) <= upper_sum (f,D2) by A5, XXREAL_0:2; ::_thesis: verum
end;
begin
theorem Th49: :: INTEGRA1:49
for A being non empty closed_interval Subset of REAL
for f being Function of A,REAL st f | A is bounded holds
upper_integral f >= lower_integral f
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f being Function of A,REAL st f | A is bounded holds
upper_integral f >= lower_integral f
let f be Function of A,REAL; ::_thesis: ( f | A is bounded implies upper_integral f >= lower_integral f )
assume A1: f | A is bounded ; ::_thesis: upper_integral f >= lower_integral f
A2: for b being real number st b in rng (upper_sum_set f) holds
lower_integral f <= b
proof
let b be real number ; ::_thesis: ( b in rng (upper_sum_set f) implies lower_integral f <= b )
assume b in rng (upper_sum_set f) ; ::_thesis: lower_integral f <= b
then consider D1 being Element of divs A such that
D1 in dom (upper_sum_set f) and
A3: b = (upper_sum_set f) . D1 by PARTFUN1:3;
reconsider D1 = D1 as Division of A by Def3;
A4: for a being real number st a in rng (lower_sum_set f) holds
a <= upper_sum (f,D1)
proof
let a be real number ; ::_thesis: ( a in rng (lower_sum_set f) implies a <= upper_sum (f,D1) )
assume a in rng (lower_sum_set f) ; ::_thesis: a <= upper_sum (f,D1)
then consider D2 being Element of divs A such that
D2 in dom (lower_sum_set f) and
A5: a = (lower_sum_set f) . D2 by PARTFUN1:3;
reconsider D2 = D2 as Division of A by Def3;
a = lower_sum (f,D2) by A5, Def11;
hence a <= upper_sum (f,D1) by A1, Th48; ::_thesis: verum
end;
b = upper_sum (f,D1) by A3, Def10;
hence lower_integral f <= b by A4, SEQ_4:45; ::_thesis: verum
end;
thus upper_integral f >= lower_integral f by A2, SEQ_4:43; ::_thesis: verum
end;
theorem Th50: :: INTEGRA1:50
for X, Y being Subset of REAL holds (-- X) ++ (-- Y) = -- (X ++ Y)
proof
let X, Y be Subset of REAL; ::_thesis: (-- X) ++ (-- Y) = -- (X ++ Y)
for z being set st z in -- (X ++ Y) holds
z in (-- X) ++ (-- Y)
proof
let z be set ; ::_thesis: ( z in -- (X ++ Y) implies z in (-- X) ++ (-- Y) )
assume A1: z in -- (X ++ Y) ; ::_thesis: z in (-- X) ++ (-- Y)
reconsider XY = X ++ Y as Subset of REAL by MEMBERED:3;
z in -- XY by A1;
then consider x being Real such that
A2: x in XY and
A3: z = - x by MEASURE6:72;
consider a, b being Real such that
A4: a in X and
A5: b in Y and
A6: x = a + b by A2, MEASURE6:21;
A7: - a in -- X by A4, MEMBER_1:11;
A8: - b in -- Y by A5, MEMBER_1:11;
z = (- a) + (- b) by A3, A6;
hence z in (-- X) ++ (-- Y) by A7, A8, MEMBER_1:46; ::_thesis: verum
end;
then A9: -- (X ++ Y) c= (-- X) ++ (-- Y) by TARSKI:def_3;
for z being set st z in (-- X) ++ (-- Y) holds
z in -- (X ++ Y)
proof
let z be set ; ::_thesis: ( z in (-- X) ++ (-- Y) implies z in -- (X ++ Y) )
assume A10: z in (-- X) ++ (-- Y) ; ::_thesis: z in -- (X ++ Y)
(-- X) ++ (-- Y) c= REAL by MEMBERED:3;
then consider x, y being Real such that
A11: x in -- X and
A12: y in -- Y and
A13: z = x + y by A10, MEASURE6:21;
consider b being Real such that
A14: b in Y and
A15: y = - b by A12, MEASURE6:72;
reconsider X = X as Subset of REAL ;
consider a being Real such that
A16: a in X and
A17: x = - a by A11, MEASURE6:72;
A18: a + b in X ++ Y by A16, A14, MEMBER_1:46;
z = - (a + b) by A13, A17, A15;
hence z in -- (X ++ Y) by A18, MEMBER_1:11; ::_thesis: verum
end;
then (-- X) ++ (-- Y) c= -- (X ++ Y) by TARSKI:def_3;
hence (-- X) ++ (-- Y) = -- (X ++ Y) by A9, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th51: :: INTEGRA1:51
for X, Y being Subset of REAL st X is bounded_above & Y is bounded_above holds
X ++ Y is bounded_above
proof
let X, Y be Subset of REAL; ::_thesis: ( X is bounded_above & Y is bounded_above implies X ++ Y is bounded_above )
assume that
A1: X is bounded_above and
A2: Y is bounded_above ; ::_thesis: X ++ Y is bounded_above
A3: -- Y is bounded_below by A2, MEASURE6:41;
-- X is bounded_below by A1, MEASURE6:41;
then A4: (-- X) ++ (-- Y) is bounded_below by A3, SEQ_4:124;
reconsider XY = X ++ Y as Subset of REAL by MEMBERED:3;
-- XY is bounded_below by Th50, A4;
hence X ++ Y is bounded_above by MEASURE6:41; ::_thesis: verum
end;
theorem Th52: :: INTEGRA1:52
for X, Y being non empty Subset of REAL st X is bounded_above & Y is bounded_above holds
upper_bound (X ++ Y) = (upper_bound X) + (upper_bound Y)
proof
let X, Y be non empty Subset of REAL; ::_thesis: ( X is bounded_above & Y is bounded_above implies upper_bound (X ++ Y) = (upper_bound X) + (upper_bound Y) )
assume that
A1: X is bounded_above and
A2: Y is bounded_above ; ::_thesis: upper_bound (X ++ Y) = (upper_bound X) + (upper_bound Y)
A3: -- Y is bounded_below by A2, MEASURE6:41;
A4: -- X is bounded_below by A1, MEASURE6:41;
then lower_bound ((-- X) ++ (-- Y)) = (lower_bound (-- X)) + (lower_bound (-- Y)) by A3, SEQ_4:125;
then A5: lower_bound ((-- X) ++ (-- Y)) = (- (upper_bound (-- (-- X)))) + (lower_bound (-- Y)) by A4, MEASURE6:43
.= (- (upper_bound X)) + (- (upper_bound (-- (-- Y)))) by A3, MEASURE6:43
.= - ((upper_bound X) + (upper_bound Y)) ;
A6: (-- X) ++ (-- Y) = -- (X ++ Y) by Th50;
then A7: -- (X ++ Y) is bounded_below by A4, A3, SEQ_4:124;
reconsider XY = X ++ Y as Subset of REAL by MEMBERED:3;
- (upper_bound (-- (-- XY))) = - ((upper_bound X) + (upper_bound Y)) by A6, A5, A7, MEASURE6:43;
then upper_bound XY = (upper_bound X) + (upper_bound Y) ;
hence upper_bound (X ++ Y) = (upper_bound X) + (upper_bound Y) ; ::_thesis: verum
end;
theorem Th53: :: INTEGRA1:53
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_above & g | A is bounded_above holds
(upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i)
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_above & g | A is bounded_above holds
(upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i)
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_above & g | A is bounded_above holds
(upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i)
let D be Division of A; ::_thesis: for f, g being Function of A,REAL st i in dom D & f | A is bounded_above & g | A is bounded_above holds
(upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i)
let f, g be Function of A,REAL; ::_thesis: ( i in dom D & f | A is bounded_above & g | A is bounded_above implies (upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i) )
assume A1: i in dom D ; ::_thesis: ( not f | A is bounded_above or not g | A is bounded_above or (upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i) )
dom (f + g) = A /\ A by FUNCT_2:def_1;
then dom ((f + g) | (divset (D,i))) = divset (D,i) by A1, Th8, RELAT_1:62;
then A2: not rng ((f + g) | (divset (D,i))) is empty by RELAT_1:42;
(f + g) | (divset (D,i)) = (f | (divset (D,i))) + (g | (divset (D,i))) by RFUNCT_1:44;
then A3: rng ((f + g) | (divset (D,i))) c= (rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i)))) by Th10;
assume f | A is bounded_above ; ::_thesis: ( not g | A is bounded_above or (upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i) )
then rng f is bounded_above by Th13;
then A4: rng (f | (divset (D,i))) is bounded_above by RELAT_1:70, XXREAL_2:43;
dom g = A by FUNCT_2:def_1;
then dom (g | (divset (D,i))) = divset (D,i) by A1, Th8, RELAT_1:62;
then A5: not rng (g | (divset (D,i))) is empty by RELAT_1:42;
A6: 0 <= vol (divset (D,i)) by SEQ_4:11, XREAL_1:48;
assume g | A is bounded_above ; ::_thesis: (upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i)
then rng g is bounded_above by Th13;
then A7: rng (g | (divset (D,i))) is bounded_above by RELAT_1:70, XXREAL_2:43;
then A8: (rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i)))) is bounded_above by A4, Th51;
dom f = A by FUNCT_2:def_1;
then dom (f | (divset (D,i))) = divset (D,i) by A1, Th8, RELAT_1:62;
then not rng (f | (divset (D,i))) is empty by RELAT_1:42;
then upper_bound ((rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i))))) = (upper_bound (rng (f | (divset (D,i))))) + (upper_bound (rng (g | (divset (D,i))))) by A4, A7, A5, Th52;
then (upper_bound (rng ((f + g) | (divset (D,i))))) * (vol (divset (D,i))) <= ((upper_bound (rng (f | (divset (D,i))))) + (upper_bound (rng (g | (divset (D,i)))))) * (vol (divset (D,i))) by A8, A2, A6, A3, SEQ_4:48, XREAL_1:64;
then (upper_volume ((f + g),D)) . i <= ((upper_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))) + ((upper_bound (rng (g | (divset (D,i))))) * (vol (divset (D,i)))) by A1, Def6;
then (upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_bound (rng (g | (divset (D,i))))) * (vol (divset (D,i)))) by A1, Def6;
hence (upper_volume ((f + g),D)) . i <= ((upper_volume (f,D)) . i) + ((upper_volume (g,D)) . i) by A1, Def6; ::_thesis: verum
end;
theorem Th54: :: INTEGRA1:54
for i being Element of NAT
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
proof
let i be Element of NAT ; ::_thesis: for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
let D be Division of A; ::_thesis: for f, g being Function of A,REAL st i in dom D & f | A is bounded_below & g | A is bounded_below holds
((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
let f, g be Function of A,REAL; ::_thesis: ( i in dom D & f | A is bounded_below & g | A is bounded_below implies ((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i )
assume that
A1: i in dom D and
A2: f | A is bounded_below and
A3: g | A is bounded_below ; ::_thesis: ((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i
A4: 0 <= vol (divset (D,i)) by SEQ_4:11, XREAL_1:48;
dom (f + g) = A /\ A by FUNCT_2:def_1;
then dom ((f + g) | (divset (D,i))) = divset (D,i) by A1, Th8, RELAT_1:62;
then A5: not rng ((f + g) | (divset (D,i))) is empty by RELAT_1:42;
rng g is bounded_below by A3, Th11;
then A6: rng (g | (divset (D,i))) is bounded_below by RELAT_1:70, XXREAL_2:44;
dom g = A by FUNCT_2:def_1;
then dom (g | (divset (D,i))) = divset (D,i) by A1, Th8, RELAT_1:62;
then A7: not rng (g | (divset (D,i))) is empty by RELAT_1:42;
(f + g) | (divset (D,i)) = (f | (divset (D,i))) + (g | (divset (D,i))) by RFUNCT_1:44;
then A8: rng ((f + g) | (divset (D,i))) c= (rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i)))) by Th10;
rng f is bounded_below by A2, Th11;
then A9: rng (f | (divset (D,i))) is bounded_below by RELAT_1:70, XXREAL_2:44;
then A10: (rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i)))) is bounded_below by A6, SEQ_4:124;
dom f = A by FUNCT_2:def_1;
then dom (f | (divset (D,i))) = divset (D,i) by A1, Th8, RELAT_1:62;
then not rng (f | (divset (D,i))) is empty by RELAT_1:42;
then lower_bound ((rng (f | (divset (D,i)))) ++ (rng (g | (divset (D,i))))) = (lower_bound (rng (f | (divset (D,i))))) + (lower_bound (rng (g | (divset (D,i))))) by A9, A6, A7, SEQ_4:125;
then (lower_bound (rng ((f + g) | (divset (D,i))))) * (vol (divset (D,i))) >= ((lower_bound (rng (f | (divset (D,i))))) + (lower_bound (rng (g | (divset (D,i)))))) * (vol (divset (D,i))) by A10, A5, A4, A8, SEQ_4:47, XREAL_1:64;
then (lower_volume ((f + g),D)) . i >= ((lower_bound (rng (f | (divset (D,i))))) * (vol (divset (D,i)))) + ((lower_bound (rng (g | (divset (D,i))))) * (vol (divset (D,i)))) by A1, Def7;
then (lower_volume ((f + g),D)) . i >= ((lower_volume (f,D)) . i) + ((lower_bound (rng (g | (divset (D,i))))) * (vol (divset (D,i)))) by A1, Def7;
hence ((lower_volume (f,D)) . i) + ((lower_volume (g,D)) . i) <= (lower_volume ((f + g),D)) . i by A1, Def7; ::_thesis: verum
end;
theorem Th55: :: INTEGRA1:55
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st f | A is bounded_above & g | A is bounded_above holds
upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D))
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f, g being Function of A,REAL st f | A is bounded_above & g | A is bounded_above holds
upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D))
let D be Division of A; ::_thesis: for f, g being Function of A,REAL st f | A is bounded_above & g | A is bounded_above holds
upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D))
let f, g be Function of A,REAL; ::_thesis: ( f | A is bounded_above & g | A is bounded_above implies upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D)) )
assume that
A1: f | A is bounded_above and
A2: g | A is bounded_above ; ::_thesis: upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D))
set H = upper_volume ((f + g),D);
set G = upper_volume (g,D);
set F = upper_volume (f,D);
len (upper_volume (g,D)) = len D by Def6;
then A3: upper_volume (g,D) is Element of (len D) -tuples_on REAL by FINSEQ_2:92;
len (upper_volume (f,D)) = len D by Def6;
then A4: upper_volume (f,D) is Element of (len D) -tuples_on REAL by FINSEQ_2:92;
A5: for j being Nat st j in Seg (len D) holds
(upper_volume ((f + g),D)) . j <= ((upper_volume (f,D)) + (upper_volume (g,D))) . j
proof
let j be Nat; ::_thesis: ( j in Seg (len D) implies (upper_volume ((f + g),D)) . j <= ((upper_volume (f,D)) + (upper_volume (g,D))) . j )
assume j in Seg (len D) ; ::_thesis: (upper_volume ((f + g),D)) . j <= ((upper_volume (f,D)) + (upper_volume (g,D))) . j
then j in dom D by FINSEQ_1:def_3;
then (upper_volume ((f + g),D)) . j <= ((upper_volume (f,D)) . j) + ((upper_volume (g,D)) . j) by A1, A2, Th53;
hence (upper_volume ((f + g),D)) . j <= ((upper_volume (f,D)) + (upper_volume (g,D))) . j by A4, A3, RVSUM_1:11; ::_thesis: verum
end;
len (upper_volume ((f + g),D)) = len D by Def6;
then A6: upper_volume ((f + g),D) is Element of (len D) -tuples_on REAL by FINSEQ_2:92;
(upper_volume (f,D)) + (upper_volume (g,D)) is Element of (len D) -tuples_on REAL by A4, A3, FINSEQ_2:120;
then Sum (upper_volume ((f + g),D)) <= Sum ((upper_volume (f,D)) + (upper_volume (g,D))) by A6, A5, RVSUM_1:82;
hence upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D)) by A4, A3, RVSUM_1:89; ::_thesis: verum
end;
theorem Th56: :: INTEGRA1:56
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f, g being Function of A,REAL st f | A is bounded_below & g | A is bounded_below holds
(lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D)
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for D being Division of A
for f, g being Function of A,REAL st f | A is bounded_below & g | A is bounded_below holds
(lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D)
let D be Division of A; ::_thesis: for f, g being Function of A,REAL st f | A is bounded_below & g | A is bounded_below holds
(lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D)
let f, g be Function of A,REAL; ::_thesis: ( f | A is bounded_below & g | A is bounded_below implies (lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D) )
assume that
A1: f | A is bounded_below and
A2: g | A is bounded_below ; ::_thesis: (lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D)
set H = lower_volume ((f + g),D);
set G = lower_volume (g,D);
set F = lower_volume (f,D);
len (lower_volume (g,D)) = len D by Def7;
then A3: lower_volume (g,D) is Element of (len D) -tuples_on REAL by FINSEQ_2:92;
len (lower_volume (f,D)) = len D by Def7;
then A4: lower_volume (f,D) is Element of (len D) -tuples_on REAL by FINSEQ_2:92;
A5: for j being Nat st j in Seg (len D) holds
((lower_volume (f,D)) + (lower_volume (g,D))) . j <= (lower_volume ((f + g),D)) . j
proof
let j be Nat; ::_thesis: ( j in Seg (len D) implies ((lower_volume (f,D)) + (lower_volume (g,D))) . j <= (lower_volume ((f + g),D)) . j )
assume j in Seg (len D) ; ::_thesis: ((lower_volume (f,D)) + (lower_volume (g,D))) . j <= (lower_volume ((f + g),D)) . j
then j in dom D by FINSEQ_1:def_3;
then ((lower_volume (f,D)) . j) + ((lower_volume (g,D)) . j) <= (lower_volume ((f + g),D)) . j by A1, A2, Th54;
hence ((lower_volume (f,D)) + (lower_volume (g,D))) . j <= (lower_volume ((f + g),D)) . j by A4, A3, RVSUM_1:11; ::_thesis: verum
end;
len (lower_volume ((f + g),D)) = len D by Def7;
then A6: lower_volume ((f + g),D) is Element of (len D) -tuples_on REAL by FINSEQ_2:92;
(lower_volume (f,D)) + (lower_volume (g,D)) is Element of (len D) -tuples_on REAL by A4, A3, FINSEQ_2:120;
then Sum ((lower_volume (f,D)) + (lower_volume (g,D))) <= Sum (lower_volume ((f + g),D)) by A6, A5, RVSUM_1:82;
hence (lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D) by A4, A3, RVSUM_1:89; ::_thesis: verum
end;
theorem :: INTEGRA1:57
for A being non empty closed_interval Subset of REAL
for f, g being Function of A,REAL st f | A is bounded & g | A is bounded & f is integrable & g is integrable holds
( f + g is integrable & integral (f + g) = (integral f) + (integral g) )
proof
let A be non empty closed_interval Subset of REAL; ::_thesis: for f, g being Function of A,REAL st f | A is bounded & g | A is bounded & f is integrable & g is integrable holds
( f + g is integrable & integral (f + g) = (integral f) + (integral g) )
let f, g be Function of A,REAL; ::_thesis: ( f | A is bounded & g | A is bounded & f is integrable & g is integrable implies ( f + g is integrable & integral (f + g) = (integral f) + (integral g) ) )
assume that
A1: f | A is bounded and
A2: g | A is bounded and
A3: f is integrable and
A4: g is integrable ; ::_thesis: ( f + g is integrable & integral (f + g) = (integral f) + (integral g) )
A5: (lower_integral f) + (lower_integral g) = (upper_integral f) + (lower_integral g) by A3, Def16
.= (integral f) + (integral g) by A4, Def16 ;
A6: (f + g) | (A /\ A) is bounded by A1, A2, RFUNCT_1:83;
for D being set st D in (divs A) /\ (dom (lower_sum_set (f + g))) holds
(lower_sum_set (f + g)) . D <= ((upper_bound (rng f)) * (vol A)) + ((upper_bound (rng g)) * (vol A))
proof
let D be set ; ::_thesis: ( D in (divs A) /\ (dom (lower_sum_set (f + g))) implies (lower_sum_set (f + g)) . D <= ((upper_bound (rng f)) * (vol A)) + ((upper_bound (rng g)) * (vol A)) )
assume D in (divs A) /\ (dom (lower_sum_set (f + g))) ; ::_thesis: (lower_sum_set (f + g)) . D <= ((upper_bound (rng f)) * (vol A)) + ((upper_bound (rng g)) * (vol A))
then reconsider D = D as Division of A by Def3;
(lower_sum_set (f + g)) . D = lower_sum ((f + g),D) by Def11;
then A7: (lower_sum_set (f + g)) . D <= upper_sum ((f + g),D) by A6, Th28;
upper_sum (f,D) <= (upper_bound (rng f)) * (vol A) by A1, Th27;
then A8: (upper_sum (f,D)) + (upper_sum (g,D)) <= ((upper_bound (rng f)) * (vol A)) + (upper_sum (g,D)) by XREAL_1:6;
upper_sum (g,D) <= (upper_bound (rng g)) * (vol A) by A2, Th27;
then A9: ((upper_bound (rng f)) * (vol A)) + (upper_sum (g,D)) <= ((upper_bound (rng f)) * (vol A)) + ((upper_bound (rng g)) * (vol A)) by XREAL_1:6;
upper_sum ((f + g),D) <= (upper_sum (f,D)) + (upper_sum (g,D)) by A1, A2, Th55;
then (lower_sum_set (f + g)) . D <= (upper_sum (f,D)) + (upper_sum (g,D)) by A7, XXREAL_0:2;
then (lower_sum_set (f + g)) . D <= ((upper_bound (rng f)) * (vol A)) + (upper_sum (g,D)) by A8, XXREAL_0:2;
hence (lower_sum_set (f + g)) . D <= ((upper_bound (rng f)) * (vol A)) + ((upper_bound (rng g)) * (vol A)) by A9, XXREAL_0:2; ::_thesis: verum
end;
then (lower_sum_set (f + g)) | (divs A) is bounded_above by RFUNCT_1:70;
then A10: rng (lower_sum_set (f + g)) is bounded_above by Th13;
then A11: f + g is lower_integrable by Def13;
for D being set st D in (divs A) /\ (dom (upper_sum_set (f + g))) holds
((lower_bound (rng f)) * (vol A)) + ((lower_bound (rng g)) * (vol A)) <= (upper_sum_set (f + g)) . D
proof
let D be set ; ::_thesis: ( D in (divs A) /\ (dom (upper_sum_set (f + g))) implies ((lower_bound (rng f)) * (vol A)) + ((lower_bound (rng g)) * (vol A)) <= (upper_sum_set (f + g)) . D )
assume D in (divs A) /\ (dom (upper_sum_set (f + g))) ; ::_thesis: ((lower_bound (rng f)) * (vol A)) + ((lower_bound (rng g)) * (vol A)) <= (upper_sum_set (f + g)) . D
then reconsider D = D as Division of A by Def3;
(upper_sum_set (f + g)) . D = upper_sum ((f + g),D) by Def10;
then A12: lower_sum ((f + g),D) <= (upper_sum_set (f + g)) . D by A6, Th28;
(lower_bound (rng f)) * (vol A) <= lower_sum (f,D) by A1, Th25;
then A13: ((lower_bound (rng f)) * (vol A)) + (lower_sum (g,D)) <= (lower_sum (f,D)) + (lower_sum (g,D)) by XREAL_1:6;
(lower_bound (rng g)) * (vol A) <= lower_sum (g,D) by A2, Th25;
then A14: ((lower_bound (rng f)) * (vol A)) + ((lower_bound (rng g)) * (vol A)) <= ((lower_bound (rng f)) * (vol A)) + (lower_sum (g,D)) by XREAL_1:6;
(lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D) by A1, A2, Th56;
then (lower_sum (f,D)) + (lower_sum (g,D)) <= (upper_sum_set (f + g)) . D by A12, XXREAL_0:2;
then ((lower_bound (rng f)) * (vol A)) + (lower_sum (g,D)) <= (upper_sum_set (f + g)) . D by A13, XXREAL_0:2;
hence ((lower_bound (rng f)) * (vol A)) + ((lower_bound (rng g)) * (vol A)) <= (upper_sum_set (f + g)) . D by A14, XXREAL_0:2; ::_thesis: verum
end;
then (upper_sum_set (f + g)) | (divs A) is bounded_below by RFUNCT_1:71;
then A15: rng (upper_sum_set (f + g)) is bounded_below by Th11;
A16: for D being Division of A st D in (divs A) /\ (dom (upper_sum_set (f + g))) holds
((upper_sum_set f) . D) + ((upper_sum_set g) . D) >= upper_integral (f + g)
proof
let D be Division of A; ::_thesis: ( D in (divs A) /\ (dom (upper_sum_set (f + g))) implies ((upper_sum_set f) . D) + ((upper_sum_set g) . D) >= upper_integral (f + g) )
(upper_sum (f,D)) + (upper_sum (g,D)) >= upper_sum ((f + g),D) by A1, A2, Th55;
then A17: ((upper_sum_set f) . D) + (upper_sum (g,D)) >= upper_sum ((f + g),D) by Def10;
assume D in (divs A) /\ (dom (upper_sum_set (f + g))) ; ::_thesis: ((upper_sum_set f) . D) + ((upper_sum_set g) . D) >= upper_integral (f + g)
then D in dom (upper_sum_set (f + g)) by XBOOLE_0:def_4;
then (upper_sum_set (f + g)) . D in rng (upper_sum_set (f + g)) by FUNCT_1:def_3;
then A18: (upper_sum_set (f + g)) . D >= upper_integral (f + g) by A15, SEQ_4:def_2;
((upper_sum_set f) . D) + ((upper_sum_set g) . D) >= upper_sum ((f + g),D) by A17, Def10;
then ((upper_sum_set f) . D) + ((upper_sum_set g) . D) >= (upper_sum_set (f + g)) . D by Def10;
hence ((upper_sum_set f) . D) + ((upper_sum_set g) . D) >= upper_integral (f + g) by A18, XXREAL_0:2; ::_thesis: verum
end;
A19: dom (upper_sum_set (f + g)) = divs A by FUNCT_2:def_1;
A20: for a1 being real number st a1 in rng (upper_sum_set f) holds
a1 >= (upper_integral (f + g)) - (upper_integral g)
proof
let a1 be real number ; ::_thesis: ( a1 in rng (upper_sum_set f) implies a1 >= (upper_integral (f + g)) - (upper_integral g) )
assume a1 in rng (upper_sum_set f) ; ::_thesis: a1 >= (upper_integral (f + g)) - (upper_integral g)
then consider D1 being Element of divs A such that
D1 in dom (upper_sum_set f) and
A21: a1 = (upper_sum_set f) . D1 by PARTFUN1:3;
reconsider D1 = D1 as Division of A by Def3;
A22: a1 = upper_sum (f,D1) by A21, Def10;
for a2 being real number st a2 in rng (upper_sum_set g) holds
a2 >= (upper_integral (f + g)) - a1
proof
let a2 be real number ; ::_thesis: ( a2 in rng (upper_sum_set g) implies a2 >= (upper_integral (f + g)) - a1 )
assume a2 in rng (upper_sum_set g) ; ::_thesis: a2 >= (upper_integral (f + g)) - a1
then consider D2 being Element of divs A such that
D2 in dom (upper_sum_set g) and
A23: a2 = (upper_sum_set g) . D2 by PARTFUN1:3;
reconsider D2 = D2 as Division of A by Def3;
consider D being Division of A such that
A24: D1 <= D and
A25: D2 <= D by Th47;
A26: D in divs A by Def3;
(upper_sum_set g) . D = upper_sum (g,D) by Def10;
then A27: (upper_sum_set g) . D <= upper_sum (g,D2) by A2, A25, Th45;
(upper_sum_set f) . D = upper_sum (f,D) by Def10;
then (upper_sum_set f) . D <= upper_sum (f,D1) by A1, A24, Th45;
then A28: ((upper_sum_set f) . D) + ((upper_sum_set g) . D) <= (upper_sum (f,D1)) + (upper_sum (g,D2)) by A27, XREAL_1:7;
((upper_sum_set f) . D) + ((upper_sum_set g) . D) >= upper_integral (f + g) by A19, A16, A26;
then A29: (upper_sum (f,D1)) + (upper_sum (g,D2)) >= upper_integral (f + g) by A28, XXREAL_0:2;
a2 = upper_sum (g,D2) by A23, Def10;
hence a2 >= (upper_integral (f + g)) - a1 by A22, A29, XREAL_1:20; ::_thesis: verum
end;
then lower_bound (rng (upper_sum_set g)) >= (upper_integral (f + g)) - a1 by SEQ_4:43;
then a1 + (lower_bound (rng (upper_sum_set g))) >= upper_integral (f + g) by XREAL_1:20;
hence a1 >= (upper_integral (f + g)) - (upper_integral g) by XREAL_1:20; ::_thesis: verum
end;
upper_integral f >= (upper_integral (f + g)) - (upper_integral g) by A20, SEQ_4:43;
then A30: (integral f) + (upper_integral g) >= upper_integral (f + g) by XREAL_1:20;
A31: for D being Division of A st D in (divs A) /\ (dom (lower_sum_set (f + g))) holds
((lower_sum_set f) . D) + ((lower_sum_set g) . D) <= lower_integral (f + g)
proof
let D be Division of A; ::_thesis: ( D in (divs A) /\ (dom (lower_sum_set (f + g))) implies ((lower_sum_set f) . D) + ((lower_sum_set g) . D) <= lower_integral (f + g) )
(lower_sum (f,D)) + (lower_sum (g,D)) <= lower_sum ((f + g),D) by A1, A2, Th56;
then A32: ((lower_sum_set f) . D) + (lower_sum (g,D)) <= lower_sum ((f + g),D) by Def11;
assume D in (divs A) /\ (dom (lower_sum_set (f + g))) ; ::_thesis: ((lower_sum_set f) . D) + ((lower_sum_set g) . D) <= lower_integral (f + g)
then D in dom (lower_sum_set (f + g)) by XBOOLE_0:def_4;
then (lower_sum_set (f + g)) . D in rng (lower_sum_set (f + g)) by FUNCT_1:def_3;
then A33: (lower_sum_set (f + g)) . D <= lower_integral (f + g) by A10, SEQ_4:def_1;
((lower_sum_set f) . D) + ((lower_sum_set g) . D) <= lower_sum ((f + g),D) by A32, Def11;
then ((lower_sum_set f) . D) + ((lower_sum_set g) . D) <= (lower_sum_set (f + g)) . D by Def11;
hence ((lower_sum_set f) . D) + ((lower_sum_set g) . D) <= lower_integral (f + g) by A33, XXREAL_0:2; ::_thesis: verum
end;
A34: dom (lower_sum_set (f + g)) = divs A by FUNCT_2:def_1;
A35: for a1 being real number st a1 in rng (lower_sum_set f) holds
a1 <= (lower_integral (f + g)) - (lower_integral g)
proof
let a1 be real number ; ::_thesis: ( a1 in rng (lower_sum_set f) implies a1 <= (lower_integral (f + g)) - (lower_integral g) )
assume a1 in rng (lower_sum_set f) ; ::_thesis: a1 <= (lower_integral (f + g)) - (lower_integral g)
then consider D1 being Element of divs A such that
D1 in dom (lower_sum_set f) and
A36: a1 = (lower_sum_set f) . D1 by PARTFUN1:3;
reconsider D1 = D1 as Division of A by Def3;
A37: a1 = lower_sum (f,D1) by A36, Def11;
for a2 being real number st a2 in rng (lower_sum_set g) holds
a2 <= (lower_integral (f + g)) - a1
proof
let a2 be real number ; ::_thesis: ( a2 in rng (lower_sum_set g) implies a2 <= (lower_integral (f + g)) - a1 )
assume a2 in rng (lower_sum_set g) ; ::_thesis: a2 <= (lower_integral (f + g)) - a1
then consider D2 being Element of divs A such that
D2 in dom (lower_sum_set g) and
A38: a2 = (lower_sum_set g) . D2 by PARTFUN1:3;
reconsider D2 = D2 as Division of A by Def3;
consider D being Division of A such that
A39: D1 <= D and
A40: D2 <= D by Th47;
A41: D in divs A by Def3;
(lower_sum_set g) . D = lower_sum (g,D) by Def11;
then A42: (lower_sum_set g) . D >= lower_sum (g,D2) by A2, A40, Th46;
(lower_sum_set f) . D = lower_sum (f,D) by Def11;
then (lower_sum_set f) . D >= lower_sum (f,D1) by A1, A39, Th46;
then A43: ((lower_sum_set f) . D) + ((lower_sum_set g) . D) >= (lower_sum (f,D1)) + (lower_sum (g,D2)) by A42, XREAL_1:7;
((lower_sum_set f) . D) + ((lower_sum_set g) . D) <= lower_integral (f + g) by A34, A31, A41;
then A44: (lower_sum (f,D1)) + (lower_sum (g,D2)) <= lower_integral (f + g) by A43, XXREAL_0:2;
a2 = lower_sum (g,D2) by A38, Def11;
hence a2 <= (lower_integral (f + g)) - a1 by A37, A44, XREAL_1:19; ::_thesis: verum
end;
then upper_bound (rng (lower_sum_set g)) <= (lower_integral (f + g)) - a1 by SEQ_4:45;
then a1 + (upper_bound (rng (lower_sum_set g))) <= lower_integral (f + g) by XREAL_1:19;
hence a1 <= (lower_integral (f + g)) - (lower_integral g) by XREAL_1:19; ::_thesis: verum
end;
upper_bound (rng (lower_sum_set f)) <= (lower_integral (f + g)) - (lower_integral g) by A35, SEQ_4:45;
then A45: (lower_integral f) + (lower_integral g) <= lower_integral (f + g) by XREAL_1:19;
A46: upper_integral (f + g) >= lower_integral (f + g) by A6, Th49;
then (integral f) + (integral g) <= upper_integral (f + g) by A45, A5, XXREAL_0:2;
then upper_integral (f + g) = (integral f) + (integral g) by A30, XXREAL_0:1;
then A47: upper_integral (f + g) = lower_integral (f + g) by A45, A46, A5, XXREAL_0:1;
f + g is upper_integrable by A15, Def12;
hence ( f + g is integrable & integral (f + g) = (integral f) + (integral g) ) by A11, A45, A5, A30, A47, Def16, XXREAL_0:1; ::_thesis: verum
end;
theorem :: INTEGRA1:58
for i, j being Element of NAT
for f being FinSequence st i in dom f & j in dom f & i <= j holds
len (mid (f,i,j)) = (j - i) + 1 by Lm1;