begin
theorem
(
- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap ((- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) holds
((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap ((- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= exp_R (- x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
r being ( ( ) (
V28()
V29()
ext-real )
Real) st
r : ( ( ) (
V28()
V29()
ext-real )
Real)
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
(#) (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (r : ( ( ) ( V28() V29() ext-real ) Real) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) holds
(((1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (r : ( ( ) ( V28() V29() ext-real ) Real) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (r : ( ( ) ( V28() V29() ext-real ) Real) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
r being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
r : ( ( ) (
V28()
V29()
ext-real )
Real)
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) holds
integral (
(exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (r : ( ( ) ( V28() V29() ext-real ) Real) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (r : ( ( ) ( V28() V29() ext-real ) Real) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (r : ( ( ) ( V28() V29() ext-real ) Real) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(- (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
(#) (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) holds
(((- (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= sin (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
n : ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) holds
integral (
(sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((- (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((- (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) st
n : ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
(#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) holds
(((1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= cos (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
n : ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) holds
integral (
(cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (AffineMap (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
integral (
((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
integral (
((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
(id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36()
continuous V49() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
(#) cos : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (x : ( ( ) ( V28() V29() ext-real ) Real) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
(- sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
+ ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(((- sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= - (x : ( ( ) ( V28() V29() ext-real ) Real) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
integral (
((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((- sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((- sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
(- cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(((- cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= - (x : ( ( ) ( V28() V29() ext-real ) Real) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
integral (
((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((- cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((- cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
integral (
(sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
integral (
((- cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
integral (
((AffineMap (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,0 : ( ( ) ( empty V21() V22() V23() V25() V26() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() V57() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (AffineMap (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,(- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (AffineMap (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ,(- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) )) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) holds
(
(1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
(#) (#Z (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( (
V6()
V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6() non
empty total V18(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) holds
(((1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (#Z (n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
#Z n : ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
begin
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
C being ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) holds
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) - g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| C : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
= (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
- (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
f1,
f2,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
C being ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
(#) (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
= ((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| C : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
f1,
f2,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
C being ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
(#) (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
= ((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| C : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
f1,
f2,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
C being ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) holds
((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
(#) (g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
= (f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
(#) ((f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || C : ( ( non empty ) ( non empty V51() V52() V53() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty ) ( non
empty V51()
V52()
V53() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
|||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= |||(g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) ;
theorem
for
f1,
f2,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
integrable &
((f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
integrable holds
|||((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= |||(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
+ |||(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
f1,
f2,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
integrable &
((f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b4 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
integrable holds
|||((f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= |||(f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
- |||(f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) holds
|||((- f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= - |||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
r being ( ( ) (
V28()
V29()
ext-real )
Real)
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) holds
|||((r : ( ( ) ( V28() V29() ext-real ) Real) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= r : ( ( ) (
V28()
V29()
ext-real )
Real)
* |||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
r,
p being ( ( ) (
V28()
V29()
ext-real )
Real)
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) holds
|||((r : ( ( ) ( V28() V29() ext-real ) Real) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,(p : ( ( ) ( V28() V29() ext-real ) Real) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= (r : ( ( ) ( V28() V29() ext-real ) Real) * p : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
* |||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
f,
g,
h being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
|||((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,h : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= |||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,(g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) h : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
((g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
|||((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= (|||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) ( V28() V29() ext-real ) Real) + (2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * |||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
+ |||(g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
begin
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
((g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
is_orthogonal_with g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
|||((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
= |||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
+ |||(g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
integrable & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
>= 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
|||(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) )||| : ( ( ) (
V28()
V29()
ext-real )
Real)
>= 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ;
begin
theorem
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b2 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
>= 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
<= ||..f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ..|| : ( ( ) (
V28()
V29()
ext-real )
Real) ;
theorem
for
f,
g being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) st
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
(g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
|| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
total &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
((g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
bounded &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
(#) g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_integrable_on A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
is_orthogonal_with g : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
>= 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) holds
((g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) (#) g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) || A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
b3 : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
>= 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
||..(f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) + g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ..|| : ( ( ) (
V28()
V29()
ext-real )
Real)
^2 : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (||..f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ..|| : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
+ (||..g : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ..|| : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
begin
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
+ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) &
dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= - (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / ((a : ( ( ) ( V28() V29() ext-real ) Real) + x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
+ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom ((- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= 1 : ( ( ) ( non
empty V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((a : ( ( ) ( V28() V29() ext-real ) Real) + x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((- 1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
- x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
<> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= 1 : ( ( ) ( non
empty V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
/ ((a : ( ( ) ( V28() V29() ext-real ) Real) - x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
+ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= 1 : ( ( ) ( non
empty V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
/ (a : ( ( ) ( V28() V29() ext-real ) Real) + x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
- a : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= 1 : ( ( ) ( non
empty V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
- x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom (- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= 1 : ( ( ) ( non
empty V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
/ (a : ( ( ) ( V28() V29() ext-real ) Real) - x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
+ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (a : ( ( ) ( V28() V29() ext-real ) Real) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
/ (a : ( ( ) ( V28() V29() ext-real ) Real) + x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (a : ( ( ) ( V28() V29() ext-real ) Real) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (a : ( ( ) ( V28() V29() ext-real ) Real) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
+ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom (((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (a : ( ( ) ( V28() V29() ext-real ) Real) - x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ (a : ( ( ) ( V28() V29() ext-real ) Real) + x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
+ a : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (x : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) + a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
- a : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (x : ( ( ) ( V28() V29() ext-real ) Real) + a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
b,
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
+ b : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((a : ( ( ) ( V28() V29() ext-real ) Real) - b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (x : ( ( ) ( V28() V29() ext-real ) Real) + a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((a : ( ( ) ( V28() V29() ext-real ) Real) - b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((a : ( ( ) ( V28() V29() ext-real ) Real) - b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
b,
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
- b : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((a : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (x : ( ( ) ( V28() V29() ext-real ) Real) + a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) - b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((a : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((a : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
b,
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
+ b : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((a : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (x : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((a : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - ((a : ( ( ) ( V28() V29() ext-real ) Real) + b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
b,
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f1,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
= ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
* f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
- b : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f1 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((b : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (x : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ (x : ( ( ) ( V28() V29() ext-real ) Real) - b : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((b : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + ((b : ( ( ) ( V28() V29() ext-real ) Real) - a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom ln : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) &
((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
n being ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
x : ( ( ) (
V28()
V29()
ext-real )
Real)
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= n : ( ( ) (
V21()
V22()
V23()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
/ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st not
0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) )
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= - (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
+ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (a : ( ( ) ( V28() V29() ext-real ) Real) + x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
- x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((- (2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom ((- (2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (a : ( ( ) ( V28() V29() ext-real ) Real) - x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((- (2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((- (2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
+ x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom (2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (a : ( ( ) ( V28() V29() ext-real ) Real) + x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
#R (- (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
a being ( ( ) (
V28()
V29()
ext-real )
Real)
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f,
f2 being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
(
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= a : ( ( ) (
V28()
V29()
ext-real )
Real)
- x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
> 0 : ( ( ) (
empty V21()
V22()
V23()
V25()
V26()
V27()
V28()
V29()
V30()
ext-real non
negative V50()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
NAT : ( ( ) ( non
empty V21()
V22()
V23()
V51()
V52()
V53()
V54()
V55()
V56()
V57() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) ) ) ) &
dom ((- 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom ((- 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= dom f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (a : ( ( ) ( V28() V29() ext-real ) Real) - x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
#R (- (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f2 : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (((- 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (((- 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V22() V23() V27() V28() V29() V30() ext-real non negative V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( non empty V21() V22() V23() V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= x : ( ( ) (
V28()
V29()
ext-real )
Real)
* (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((((- (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() continuous V49() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom sec : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;
theorem
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
c= dom (- cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) holds
(
- cosec : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
is_differentiable_on Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
((- cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) )
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) ) ;
theorem
for
A being ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
for
f being ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
for
Z being ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) st
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) )
c= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) &
dom (- cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) )
= Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) & ( for
x being ( ( ) (
V28()
V29()
ext-real )
Real) st
x : ( ( ) (
V28()
V29()
ext-real )
Real)
in Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) ) holds
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
. x : ( ( ) (
V28()
V29()
ext-real )
Real) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
/ ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() non empty total V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ^2) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ) &
Z : ( (
open ) (
V51()
V52()
V53()
open )
Subset of ( ( ) ( )
set ) )
= dom f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) : ( ( ) (
V51()
V52()
V53() )
Element of
K19(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) ( )
set ) ) &
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,)
| A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
Element of
K19(
K20(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ,
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V34()
V35()
V36() )
set ) ) : ( ( ) ( )
set ) ) is
continuous holds
integral (
f : ( (
V6() ) (
V1()
V4(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V5(
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
V6()
V34()
V35()
V36() )
PartFunc of ,) ,
A : ( ( non
empty closed_interval ) ( non
empty V51()
V52()
V53()
closed_interval V87() )
Subset of ( ( ) ( )
set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
= ((- cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) )
- ((- cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) : ( ( ) ( V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V87() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V73() ) set ) ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) : ( ( ) (
V28()
V29()
ext-real )
Element of
REAL : ( ( ) ( non
empty V51()
V52()
V53()
V57()
V73() )
set ) ) ;