:: FDIFF_11 semantic presentation

begin

theorem :: FDIFF_11:1
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:2
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:3
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:4
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:5
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: FDIFF_11:6
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:7
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:8
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: FDIFF_11:9
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:10
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:11
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:12
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:13
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:14
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:15
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:16
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:17
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:18
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:19
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:20
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:21
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:22
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:23
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:24
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:25
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:26
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:27
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:28
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:29
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:30
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - ((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:31
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:32
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - ((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:33
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:34
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:35
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:36
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- (((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:37
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: FDIFF_11:38
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:39
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:40
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:41
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:42
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:43
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:44
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (tan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:45
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:46
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:47
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:48
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (sec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:49
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:50
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (- ((((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / ((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cosec : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:51
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:52
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:53
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:54
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:55
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: FDIFF_11:56
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:57
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: FDIFF_11:58
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:59
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) ;

theorem :: FDIFF_11:60
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:61
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:62
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:63
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - (x : ( ( ) ( V11() V12() ext-real ) Real) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:64
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:65
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for g being ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) st not 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) - ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:66
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for g being ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) st not 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > - 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) & ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) < 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((g : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) PartFunc of ,) (#) (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) + ((x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:67
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) <> 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:68
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:69
for n being ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) <> 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #Z (n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:70
for n being ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#Z n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V33( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) , REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #Z (n : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) * (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:71
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (- (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:72
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (- (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:73
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) > 0 : ( ( ) ( V10() V11() V12() ext-real V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = ((arctan : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;

theorem :: FDIFF_11:74
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real non positive negative V60() ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ,1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) holds
( (2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) (#) ((#R (3 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) = - (((arccot : ( ( V24() ) ( V19() V22( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V23( REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) V24() V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ,REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) #R (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V1() V11() V12() ext-real positive non negative ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) / (1 : ( ( ) ( V1() V10() V11() V12() ext-real positive non negative V46() V47() V48() V49() V50() V51() V60() V61() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() ) Element of K6(REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V1() V46() V47() V48() V52() V62() ) set ) ) ) ) ;