:: FDIFF_8 semantic presentation

begin

theorem :: FDIFF_8:1
for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in dom tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: FDIFF_8:2
for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in dom cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: FDIFF_8:3
for n being ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for 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 (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 ) ) : ( ( ) ( V49() V50() V51() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) holds
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 ,) / 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) = ((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 ) ) #Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) 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 ,) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) #Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( ) ( 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_8:4
for a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for 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 (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 ) ) : ( ( ) ( 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
( 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 ) ) = 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 ) ) & 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 ,) . 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) - b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) 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 ,) / 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 ) ) 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
((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 ) ) `| 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) ) : ( ( ) ( 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 ) ) / ((x : ( ( ) ( V22() V23() ext-real ) Real) - 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 ) ) ) ) ;

theorem :: FDIFF_8:5
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & 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 ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( 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 ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_8:6
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 (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) ) holds
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) / ((cos : ( ( V6() V30( 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() V30( 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 ) ) . ((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 ) ) ) : ( ( ) ( 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_8:7
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 (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) ) holds
( cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) / ((sin : ( ( V6() V30( 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() V30( 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 ) ) . ((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 ) ) ) : ( ( ) ( 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_8:8
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) * ((cos : ( ( V6() V30( 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() V30( 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 ) ) . (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) ) : ( ( ) ( 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_8:9
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) * ((sin : ( ( V6() V30( 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() V30( 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 ) ) . (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) ) : ( ( ) ( 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_8:10
for c, a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for 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 (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (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 ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) & 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( 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() V30( 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) + (b : ( ( ) ( 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) holds
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (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 ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (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 ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) = (b : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( 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) ) : ( ( ) ( 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() V30( 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() V30( 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 ) ) . ((a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( 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 ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (c : ( ( ) ( V22() V23() ext-real ) Real) * (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 ) ) ^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_8:11
for c, a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for 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 (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (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 ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) & 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( 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() V30( 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) + (b : ( ( ) ( 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) ) holds
( cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (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 ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (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 ,) + (c : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) = - ((b : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( 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) ) : ( ( ) ( 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() V30( 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() V30( 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 ) ) . ((a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( 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 ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) + (c : ( ( ) ( V22() V23() ext-real ) Real) * (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 ) ) ^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_8:12
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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 ) ) ^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_8:13
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
( cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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 ) ) ^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_8:14
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) holds
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) = 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) * ((cos : ( ( V6() V30( 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() V30( 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 ) ) ^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_8:15
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) holds
( cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) = - (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) * ((sin : ( ( V6() V30( 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() V30( 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 ) ) ^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_8:16
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() V30( 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() V30( 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) . (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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_8:17
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() V30( 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() V30( 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) . (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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_8:18
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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( V6() V30( 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() V30( 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() V30( 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() V30( 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_8:19
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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( V6() V30( 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() V30( 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() V30( 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() V30( 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_8:20
for n being ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat)
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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( 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() V30( 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) & 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) holds
( (#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( 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() V30( 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( 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() V30( 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 ) ) * tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) * ((sin : ( ( V6() V30( 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() V30( 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) - 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) / ((cos : ( ( V6() V30( 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() V30( 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) + 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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_8:21
for n being ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat)
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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( 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() V30( 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) & 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) holds
( (#Z n : ( ( V21() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( 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() V30( 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) ) : ( ( V6() V30( 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() V30( 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 ) ) * cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) * ((cos : ( ( V6() V30( 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() V30( 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) - 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) / ((sin : ( ( V6() V30( 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() V30( 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() ) ( V15() V16() V17() V21() V23() ext-real V68() ) Nat) + 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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_8:22
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
( 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (sin : ( ( V6() V30( 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() V30( 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 ) ) <> 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (sin : ( ( V6() V30( 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() V30( 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 ) ) <> 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) = 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (sin : ( ( V6() V30( 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() V30( 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_8:23
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
( 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (sin : ( ( V6() V30( 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() V30( 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 ) ) <> 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (sin : ( ( V6() V30( 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() V30( 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 ) ) <> 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) = 1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (sin : ( ( V6() V30( 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() V30( 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_8:24
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (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 ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
((tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) / ((cos : ( ( V6() V30( 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() V30( 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_8:25
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( (- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
(((- cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) / ((sin : ( ( V6() V30( 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() V30( 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_8:26
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 (((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) (#) (tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) . (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 ) ) ) : ( ( ) ( 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 ) ) / ((cos : ( ( V6() V30( 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() V30( 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 ) ) . (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 ) ) ) : ( ( ) ( 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_8:27
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 (((- (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) ) : ( ( ) ( 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 ) ) (#) (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((- (1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) ) : ( ( ) ( 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 ) ) (#) (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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) ) : ( ( ) ( 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 ) ) (#) (cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) - (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 ) ) ) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) . (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 ) ) ) : ( ( ) ( 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 ) ) / ((sin : ( ( V6() V30( 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() V30( 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 ) ) . (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 ) ) ) : ( ( ) ( 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_8:28
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 ,) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) = (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 ,) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ,) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) * (sin : ( ( V6() V30( 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() V30( 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() V30( 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() V30( 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 ) ) + (((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 ) ) / ((cos : ( ( V6() V30( 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() V30( 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_8:29
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 ,) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) = (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 ,) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ,) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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) * (cos : ( ( V6() V30( 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() V30( 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() V30( 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() V30( 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 ) ) - (((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 ) ) / ((sin : ( ( V6() V30( 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() V30( 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_8:30
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() V30( 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() V30( 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 ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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_8:31
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() V30( 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() V30( 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 ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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 ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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() V30( 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_8:32
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 ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( 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 ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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 ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) 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 ) ) / ((cos : ( ( V6() V30( 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() V30( 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_8:33
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 ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( 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 ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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 ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) 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 ) ) / ((sin : ( ( V6() V30( 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() V30( 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_8:34
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & 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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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 ) ) / (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 ) ) + ((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) / ((cos : ( ( V6() V30( 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() V30( 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_8:35
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V11() V15() V16() V17() V19() V20() V21() V22() V23() ext-real non positive non negative V49() V50() V51() V52() V53() V54() V55() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & 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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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
( ((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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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 ) ) ^) : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V6() ) ( V1() V4( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V5( REAL : ( ( ) ( V11() V49() V50() V51() V55() V70() ) set ) ) V6() V39() V40() V41() ) Element 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() V30( 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() V30( 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() V30( 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() V30( 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 ) ) / (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 ) ) - ((1 : ( ( ) ( V11() V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( V11() V15() V16() V17() 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 ) ) / ((sin : ( ( V6() V30( 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() V30( 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 ) ) ) ) ;