:: INTEGR14 semantic presentation

begin

theorem :: INTEGR14:1
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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
((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:2
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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:3
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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) * ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:4
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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
((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:5
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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
((- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:6
for n being ( ( V27() ) ( V27() V29() V30() ext-real ) Nat)
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) holds
( - ((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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
((- ((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = (n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) #Z (n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) + 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:7
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:8
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b1 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:9
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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:10
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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
((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:11
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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:12
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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
((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:13
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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
((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

theorem :: INTEGR14:14
for Z being ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( - (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() 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() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ) ;

begin

theorem :: INTEGR14:15
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:16
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:17
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:18
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:19
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) * ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:20
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) * ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:21
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) = (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:22
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) = (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:23
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) 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 V76() ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:24
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) 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 V76() ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & (- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral ((- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:25
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) 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 V76() ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:26
for n being ( ( V27() ) ( V27() V29() V30() ext-real ) Nat)
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) #Z (n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) + 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b2 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = (((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:27
for n being ( ( V27() ) ( V27() V29() V30() ext-real ) Nat)
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) #Z (n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) + 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V28() V29() V30() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b2 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- ((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- ((#Z n : ( ( V27() ) ( V27() V29() V30() ext-real ) Nat) ) : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:28
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:29
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((exp_R : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:30
for a being ( ( ) ( V28() V29() ext-real ) Real)
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) 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() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) 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() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) 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() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) 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() V62() ) set ) ) & a : ( ( ) ( V28() V29() ext-real ) Real) <> 0 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b2 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:31
for a being ( ( ) ( V28() V29() ext-real ) Real)
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) 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() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) 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() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) 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() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) 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() V62() ) set ) ) & a : ( ( ) ( V28() V29() ext-real ) Real) <> 0 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b2 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((((- (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) (#) (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b5 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:32
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:33
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:34
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:35
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (x : ( ( ) ( V28() V29() ext-real ) Real) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (((id Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5(b3 : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:36
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:37
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:38
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:39
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:40
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:41
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:42
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:43
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:44
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:45
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:46
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:47
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:48
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:49
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:50
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:51
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:52
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) - cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:53
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - (1 : ( ( ) ( V27() V28() V29() V30() ext-real V50() V51() V52() V53() V54() V55() V56() ) Element of NAT : ( ( ) ( V51() V52() V53() V54() V55() V56() V57() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) + cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:54
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:55
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:56
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:57
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) * (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:58
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((- (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((- (cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;

theorem :: INTEGR14:59
for A being ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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 V76() ) 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() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) 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() V62() ) set ) ) = (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) + ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) / ((cos : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V28() V29() ext-real ) Real) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ^2) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( V51() V52() V53() open ) Subset of ( ( ) ( ) set ) ) = dom f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) : ( ( ) ( V51() V52() V53() ) Element of K19(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4(b1 : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) = ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) - ((sin : ( ( V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V18( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) , REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V5( REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) V6() V34() V35() V36() ) Element of K19(K20(REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ,REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V1() V34() V35() V36() ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V51() V52() V53() closed_interval V76() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) : ( ( ) ( V28() V29() ext-real ) Element of REAL : ( ( ) ( non empty V51() V52() V53() V57() V62() ) set ) ) ;