:: FDIFF_7 semantic presentation

begin

theorem :: FDIFF_7:1
for x being ( ( ) ( V22() V23() ext-real ) Real) holds x : ( ( ) ( V22() V23() ext-real ) Real) #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ;

theorem :: FDIFF_7:2
for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
x : ( ( ) ( V22() V23() ext-real ) Real) #R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = sqrt x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ;

theorem :: FDIFF_7:3
for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
x : ( ( ) ( V22() V23() ext-real ) Real) #R (- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ;

theorem :: FDIFF_7:4
for r being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (r : ( ( ) ( V22() V23() ext-real ) Real) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( r : ( ( ) ( V22() V23() ext-real ) Real) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V22() V23() ext-real ) Real) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = r : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:5
for r being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (r : ( ( ) ( V22() V23() ext-real ) Real) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( r : ( ( ) ( V22() V23() ext-real ) Real) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V22() V23() ext-real ) Real) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (r : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:6
for x being ( ( ) ( V22() V23() ext-real ) Real)
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V22() V23() ext-real ) Real) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V22() V23() ext-real ) Real) & diff ((arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (diff (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ,x : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ;

theorem :: FDIFF_7:7
for x being ( ( ) ( V22() V23() ext-real ) Real)
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V22() V23() ext-real ) Real) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V22() V23() ext-real ) Real) & diff ((arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((diff (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ,x : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ;

theorem :: FDIFF_7:8
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:9
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:10
for n being ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * ((arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:11
for n being ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * ((arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:12
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:13
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:14
for a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:15
for a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (a : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:16
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:17
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:18
for a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) * (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:19
for a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) * (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - (((a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:20
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:21
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:22
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (x : ( ( ) ( V22() V23() ext-real ) Real) * ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #R (- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:23
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:24
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ,1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:25
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) / (a : ( ( ) ( V22() V23() ext-real ) Real) * (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:26
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) / (a : ( ( ) ( V22() V23() ext-real ) Real) * (sqrt (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - ((x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:27
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (x : ( ( ) ( V22() V23() ext-real ) Real) * (((a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #R (- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:28
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f3, f, f1, f2 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = arcsin : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:29
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f3, f, f1, f2 being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f1 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f3 : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((#R (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = arccos : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() V23() ext-real ) Real) / a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:30
for n being ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((- (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:31
for n being ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:32
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
x : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:33
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
x : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (ln : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:34
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:35
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:36
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:37
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:38
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:39
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:40
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:41
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:42
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:43
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:44
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:45
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) * ((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) - (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:46
for x being ( ( ) ( V22() V23() ext-real ) Real) st cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V22() V23() ext-real ) Real) & diff ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ;

theorem :: FDIFF_7:47
for x being ( ( ) ( V22() V23() ext-real ) Real) st sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V22() V23() ext-real ) Real) & diff ((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ;

theorem :: FDIFF_7:48
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z 3 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:49
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - ((2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) / ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z 3 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:50
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_7:51
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / 2 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) & sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / sin : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = - (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V11() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (cos : ( ( V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V33( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;