:: FDIFF_9 semantic presentation

begin

definition
func sec -> ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) equals :: FDIFF_9:def 1
cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ;
func cosec -> ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) equals :: FDIFF_9:def 2
sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

theorem :: FDIFF_9:1
for x being ( ( ) ( V11() V12() ext-real ) Real) st cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) <> 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V11() V12() ext-real ) Real) & diff (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ,x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ;

theorem :: FDIFF_9:2
for x being ( ( ) ( V11() V12() ext-real ) Real) st sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) <> 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V11() V12() ext-real ) Real) & diff (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ,x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ;

theorem :: FDIFF_9:3
for x being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) holds (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) #Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = 1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) #Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ;

theorem :: FDIFF_9:4
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
(sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:5
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
(cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:6
for a, b being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (a : ( ( ) ( V11() V12() ext-real ) Real) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:7
for a, b being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - ((a : ( ( ) ( V11() V12() ext-real ) Real) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:8
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:9
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:10
for c, a, b being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) = #Z 2 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((b : ( ( ) ( V11() V12() ext-real ) Real) + ((2 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:11
for c, a, b being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) = #Z 2 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * (f1 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) + (c : ( ( ) ( V11() V12() ext-real ) Real) (#) f2 : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - (((b : ( ( ) ( V11() V12() ext-real ) Real) + ((2 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . ((a : ( ( ) ( V11() V12() ext-real ) Real) + (b : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (c : ( ( ) ( V11() V12() ext-real ) Real) * (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:12
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:13
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:14
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) * ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:15
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) * ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:16
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:17
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:18
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:19
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:20
for n being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) holds
( (#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) #Z (n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) + 1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:21
for n being ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & 1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) holds
( (#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) ) : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) * cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - ((n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) #Z (n : ( ( V10() ) ( V4() V5() V6() V10() V11() V12() ext-real V47() ) Nat) + 1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:22
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:23
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom ((- cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (- cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
(((- cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:24
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:25
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - (((exp_R : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:26
for a being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) & a : ( ( ) ( V11() V12() ext-real ) Real) <> 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:27
for a being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (((- (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) & a : ( ( ) ( V11() V12() ext-real ) Real) <> 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((- (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((((- (1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / a : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) (#) (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:28
for a, b being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) holds
( f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (a : ( ( ) ( V11() V12() ext-real ) Real) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + ((((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:29
for a, b being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) holds
( f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (a : ( ( ) ( V11() V12() ext-real ) Real) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - ((((a : ( ( ) ( V11() V12() ext-real ) Real) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + b : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:30
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:31
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - (((ln : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:32
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:33
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V1() V4() V5() V6() V8() V9() V10() V11() V12() ext-real non positive non negative V47() V62() V63() V64() V65() V66() V67() V68() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - (((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:34
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:35
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - (((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:36
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - (((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:37
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:38
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:39
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - (((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:40
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = - (((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:41
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) * cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:42
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:43
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) sec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) + (((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:44
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - (((tan : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;

theorem :: FDIFF_9:45
for Z being ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V62() V63() V64() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) (#) cosec : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) PartFunc of ,) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V62() V63() V64() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) = (- ((1 : ( ( ) ( V1() V4() V5() V6() V10() V11() V12() ext-real positive non negative V47() V62() V63() V64() V65() V66() V67() V69() ) Element of NAT : ( ( ) ( V1() V4() V5() V6() V62() V63() V64() V65() V66() V67() V68() ) Element of K6(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / (sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) - (((cot : ( ( V18() ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) * (cos : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) / ((sin : ( ( V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ( V13() V16( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V17( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V18() V27( REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) , REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) V52() V53() V54() ) Element of K6(K7(REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ,REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V52() V53() V54() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V62() V63() V64() V68() V70() ) set ) ) ) ) ;