:: FDIFF_6 semantic presentation

begin

theorem :: FDIFF_6:1
for a, x being ( ( ) ( V22() V23() ext-real ) Real) st a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ;

theorem :: FDIFF_6:2
for a, x being ( ( ) ( V22() V23() ext-real ) Real) st a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (- (x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) #R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ;

theorem :: FDIFF_6:3
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:4
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (((a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:5
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((a : ( ( ) ( V22() V23() ext-real ) Real) |^ 4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:6
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (a : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / ((a : ( ( ) ( V22() V23() ext-real ) Real) |^ 4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:7
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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
( f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:8
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) & f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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
( f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) * (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:9
for n being ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
x : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:10
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f2, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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
( f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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
(((f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (x : ( ( ) ( V22() V23() ext-real ) Real) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:11
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:12
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) - (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:13
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:14
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) holds
( exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:15
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / (exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:16
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - (x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) #R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:17
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - (x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) + (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((- (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:18
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / ((log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:19
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:20
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:21
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:22
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (- (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( - (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:23
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:24
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:25
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:26
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:27
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:28
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:29
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:30
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:31
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:32
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:33
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / ((#Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:34
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:35
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / ((exp_R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) - (exp_R (- x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:36
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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
(((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:37
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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
(((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,a : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (exp_R : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (a : ( ( ) ( V22() V23() ext-real ) Real) #R x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:38
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) holds
( (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = sin (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:39
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) holds
( 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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
((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:40
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) < 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) holds
( (- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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
(((- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * (f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:41
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:42
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f, f1 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) (#) cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) + (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:43
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (cos ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:44
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (sin (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:45
for a being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / (4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (cos (a : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ^2 : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:46
for n being ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty 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 : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = - (((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V68() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:47
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) |^ 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:48
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) / 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) |^ 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:49
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_6:50
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (- (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( - (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((- (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) = (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (log (number_e : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ,x : ( ( ) ( V22() V23() ext-real ) Real) )) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;