INTEGRA8

:: INTEGRA8 semantic presentation

begin

for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) holds sin (x : ( ( ) ( V21() V29() ext-real ) Real) + ((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = sin x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:2

for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) holds sin (x : ( ( ) ( V21() V29() ext-real ) Real) + (((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (sin x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:3

for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) holds cos (x : ( ( ) ( V21() V29() ext-real ) Real) + ((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = cos x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:4

for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) holds cos (x : ( ( ) ( V21() V29() ext-real ) Real) + (((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (cos x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:5

for x being ( ( ) ( V21() V29() ext-real ) Real) st sin (x : ( ( ) ( V21() V29() ext-real ) Real) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) >= 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) holds
sin (x : ( ( ) ( V21() V29() ext-real ) Real) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = sqrt ((1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) - (cos x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:6

for x being ( ( ) ( V21() V29() ext-real ) Real) st sin (x : ( ( ) ( V21() V29() ext-real ) Real) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) < 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) holds
sin (x : ( ( ) ( V21() V29() ext-real ) Real) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (sqrt ((1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) - (cos x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:7

sin (PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:8

sin (- (PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:9

[.(- ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: INTEGRA8:10

arcsin ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:11

arcsin (- ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:12

for x being ( ( ) ( V21() V29() ext-real ) Real) st cos (x : ( ( ) ( V21() V29() ext-real ) Real) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) >= 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cos (x : ( ( ) ( V21() V29() ext-real ) Real) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = sqrt ((1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + (cos x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:13

cos (PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:14

cos ((3 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:15

arccos ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:16

arccos (- ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (3 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 4 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:17

sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:18

theorem :: INTEGRA8:19

cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:20

for L1 being ( ( V6() linear ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued linear ) LinearFunc) holds - L1 : ( ( V6() linear ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued linear ) LinearFunc) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is ( ( V6() linear ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued linear ) LinearFunc) ;

theorem :: INTEGRA8:21

for R1 being ( ( V6() RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) holds - R1 : ( ( V6() RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is ( ( V6() RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) ;

theorem :: INTEGRA8:22

theorem :: INTEGRA8:23

for f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) c= dom (- f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) holds
( - f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) st x : ( ( ) ( V21() V29() ext-real ) Real) in Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) holds
((- f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (diff (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,x : ( ( ) ( V21() V29() ext-real ) Real) )) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) ;

theorem :: INTEGRA8:24

- sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ;

theorem :: INTEGRA8:25

for x being ( ( ) ( V21() V29() ext-real ) Real) holds
( - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V21() V29() ext-real ) Real) & diff ((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ;

theorem :: INTEGRA8:26

( - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) st x : ( ( ) ( V21() V29() ext-real ) Real) in REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) holds
diff ((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) ;

theorem :: INTEGRA8:27

sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: INTEGRA8:28

cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = - sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: INTEGRA8:29

theorem :: INTEGRA8:30

sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: INTEGRA8:31

cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: INTEGRA8:32

exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: INTEGRA8:33

for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) c= dom tan : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( tan : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) st x : ( ( ) ( V21() V29() ext-real ) Real) in Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) holds
(tan : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) ;

theorem :: INTEGRA8:34

for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) c= dom cot : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( cot : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) st x : ( ( ) ( V21() V29() ext-real ) Real) in Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) holds
(cot : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) ;

theorem :: INTEGRA8:35

for r being ( ( ) ( V21() V29() ext-real ) Real) holds rng (REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) --> r : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) set ) c= REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ;

definition

let r be ( ( ) ( V21() V29() ext-real ) Real) ;

func Cst r -> ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) , REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) equals :: INTEGRA8:def 1
REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) --> r : ( ( V50() ) ( V50() ) set ) : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

end;

theorem :: INTEGRA8:36

for a, b being ( ( ) ( V21() V29() ext-real ) Real)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds chi (A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like b₃ : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(b₃ : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = (Cst 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) , REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: INTEGRA8:37

begin

theorem :: INTEGRA8:38

for a, b being ( ( ) ( V21() V29() ext-real ) Real) st a : ( ( ) ( V21() V29() ext-real ) Real) <= b : ( ( ) ( V21() V29() ext-real ) Real) holds
integral ((Cst 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) , REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,a : ( ( ) ( V21() V29() ext-real ) Real) ,b : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = b : ( ( ) ( V21() V29() ext-real ) Real) - a : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:39

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:40

theorem :: INTEGRA8:41

theorem :: INTEGRA8:42

theorem :: INTEGRA8:43

theorem :: INTEGRA8:44

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:45

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.(x : ( ( ) ( V21() V29() ext-real ) Real) + ((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(x : ( ( ) ( V21() V29() ext-real ) Real) + (((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:46

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((- sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:47

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((- sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:48

theorem :: INTEGRA8:49

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,((PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * 3 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((- sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:50

theorem :: INTEGRA8:51

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((- sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:52

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.(x : ( ( ) ( V21() V29() ext-real ) Real) + ((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(x : ( ( ) ( V21() V29() ext-real ) Real) + (((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((- sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:53

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral (exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:54

theorem :: INTEGRA8:55

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:56

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:57

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:58

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:59

for f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & dom tan : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & dom tan : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) st x : ( ( ) ( V21() V29() ext-real ) Real) in Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) holds
( f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) & cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) <> 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (tan : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (tan : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:60

for f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & dom cot : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & dom cot : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) st x : ( ( ) ( V21() V29() ext-real ) Real) in Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) holds
( f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) & sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) <> 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (cot : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (cot : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:61

theorem :: INTEGRA8:62

for f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & dom (arcsin : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) holds
( x : ( ( ) ( V21() V29() ext-real ) Real) in ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V21() V29() ext-real ) Real) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (arcsin : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (arcsin : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:63

for f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & dom (arccos : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) holds
( x : ( ( ) ( V21() V29() ext-real ) Real) in ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V21() V29() ext-real ) Real) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (arccos : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (arccos : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:64

for f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.(- ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & dom (arcsin : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) holds
( x : ( ( ) ( V21() V29() ext-real ) Real) in ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V21() V29() ext-real ) Real) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:65

for f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.(- ((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,((sqrt 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & dom (arccos : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) `| ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = dom f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V21() V29() ext-real ) Real) holds
( x : ( ( ) ( V21() V29() ext-real ) Real) in ].(- 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() V30() ext-real non positive ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( V49() V50() V51() V86() V87() V91() open ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . x : ( ( ) ( V21() V29() ext-real ) Real) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V21() V29() ext-real ) Real) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is continuous holds
integral (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = - (PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:66

for f, g being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) & (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded & g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) & (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded holds
integral (((f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (((f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) + (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:67

for f, g being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) & (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded & g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) & (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded holds
integral (((f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:68

for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,)
for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V21() V29() ext-real ) Real)
for Z being ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) st f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) c= Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_integrable_on A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) & (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) | A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is bounded holds
integral ((r : ( ( ) ( V21() V29() ext-real ) Real) (#) (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) `| Z : ( ( open ) ( V49() V50() V51() open ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (r : ( ( ) ( V21() V29() ext-real ) Real) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (r : ( ( ) ( V21() V29() ext-real ) Real) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() complex-valued ext-real-valued real-valued ) PartFunc of ,) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:69

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) + (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:70

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:71

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:72

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,((PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * 3 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:73

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:74

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:75

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.(x : ( ( ) ( V21() V29() ext-real ) Real) + ((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(x : ( ( ) ( V21() V29() ext-real ) Real) + (((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:76

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (((cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) + (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:77

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = number_e : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:78

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:79

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:80

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:81

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,((PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * 3 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:82

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:83

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:84

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.(x : ( ( ) ( V21() V29() ext-real ) Real) + ((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(x : ( ( ) ( V21() V29() ext-real ) Real) + (((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) + (2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin x : ( ( ) ( V21() V29() ext-real ) Real) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:85

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V21() V29() ext-real ) Real) holds integral ((r : ( ( ) ( V21() V29() ext-real ) Real) (#) sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (r : ( ( ) ( V21() V29() ext-real ) Real) * ((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (r : ( ( ) ( V21() V29() ext-real ) Real) * ((- cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:86

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V21() V29() ext-real ) Real) holds integral ((r : ( ( ) ( V21() V29() ext-real ) Real) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (r : ( ( ) ( V21() V29() ext-real ) Real) * (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (r : ( ( ) ( V21() V29() ext-real ) Real) * (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:87

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V21() V29() ext-real ) Real) holds integral ((r : ( ( ) ( V21() V29() ext-real ) Real) (#) sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (r : ( ( ) ( V21() V29() ext-real ) Real) * (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (r : ( ( ) ( V21() V29() ext-real ) Real) * (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:88

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V21() V29() ext-real ) Real) holds integral ((r : ( ( ) ( V21() V29() ext-real ) Real) (#) cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (r : ( ( ) ( V21() V29() ext-real ) Real) * (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (r : ( ( ) ( V21() V29() ext-real ) Real) * (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:89

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for r being ( ( ) ( V21() V29() ext-real ) Real) holds integral ((r : ( ( ) ( V21() V29() ext-real ) Real) (#) exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (r : ( ( ) ( V21() V29() ext-real ) Real) * (exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (r : ( ( ) ( V21() V29() ext-real ) Real) * (exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:90

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real non negative ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (((cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:91

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real non negative ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:92

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:93

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (3 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real non negative ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real non negative ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:94

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ,(PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:95

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:96

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) )
for x being ( ( ) ( V21() V29() ext-real ) Real)
for n being ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) st A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) = [.(x : ( ( ) ( V21() V29() ext-real ) Real) + ((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ,(x : ( ( ) ( V21() V29() ext-real ) Real) + (((2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * n : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V28() V29() V30() V31() ext-real V49() V50() V51() V52() V53() V54() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) .] : ( ( ) ( V49() V50() V51() V91() closed ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) holds
integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = 0 : ( ( ) ( empty V21() V28() V29() V30() V31() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V88() V91() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: INTEGRA8:97

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (((cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) + (integral ((cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) )) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:98

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (((cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - (integral ((cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) )) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:99

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((sinh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real non negative ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (((cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (cosh : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:100

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = (1 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V21() V28() V29() V30() V31() ext-real positive non negative V49() V50() V51() V52() V53() V54() V86() V88() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() V88() ) Element of K19(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V21() V29() ext-real non negative ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) * (((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ^2) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:101

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;

theorem :: INTEGRA8:102

for A being ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) holds integral ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) - sin : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) = ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (upper_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) - ((exp_R : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() quasi_total ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -defined REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) -valued V6() non empty total quasi_total complex-valued ext-real-valued real-valued continuous ) Element of K19(K20(REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ,REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) . (lower_bound A : ( ( non empty closed_interval ) ( non empty V49() V50() V51() closed_interval V88() V89() V90() V91() compact closed ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) : ( ( ) ( V21() V29() ext-real ) Element of REAL : ( ( ) ( non empty V34() V49() V50() V51() V55() V88() V89() V91() ) set ) ) ;