:: FDIFF_4 semantic presentation

begin

theorem :: FDIFF_4:1
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 (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 ) & ( 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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) + 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_4:2
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 (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 ) & ( 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) - a : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - 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_4:3
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 (- (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 ) & ( 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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - 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_4:4
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 ((id 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 ) ) - (a : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ,) = 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (id 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 ) ) - (a : ( ( ) ( V22() V23() ext-real ) Real) (#) 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
(((id 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 ) ) - (a : ( ( ) ( V22() V23() ext-real ) Real) (#) 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 ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) / (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 ) ) ) ) ;

theorem :: FDIFF_4:5
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) ) - (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) ) - (id 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 ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) ) - (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - 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) + 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_4:6
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 ((id 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 ) ) - ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ,) = 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (id 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 ) ) - ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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
(((id 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 ) ) - ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) ) = (x : ( ( ) ( V22() V23() ext-real ) Real) - 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) + 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_4:7
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 ((id 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 ) ) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ,) = 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (id 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 ) ) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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
(((id 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 ) ) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) 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 ) ) = (x : ( ( ) ( V22() V23() ext-real ) Real) + 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) - 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_4:8
for a, b 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 ((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) = 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( 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_4:9
for a, b 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 ((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) = 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(((id 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 ) ) + ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( 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_4:10
for a, b 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 ((id 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 ) ) - ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) = 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (id 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 ) ) - ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(((id 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 ) ) - ((a : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) + b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( 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_4:11
for b, 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 ((id 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 ) ) + ((b : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) = 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (id 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 ) ) + ((b : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(((id 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 ) ) + ((b : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( 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_4:12
for c, a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non 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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( 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 ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = b : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( 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 ) ) ) ) ;

theorem :: FDIFF_4:13
for c, a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non 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 ) ) * (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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) + (b : ( ( ) ( 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 ) ) & (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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) * (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 ,) + (c : ( ( ) ( 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = (b : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * c : ( ( ) ( 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 ) ) / ((a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) + (c : ( ( ) ( V22() V23() ext-real ) Real) * (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_4:14
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 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 ) & ( 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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) 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 ,) ^ : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) + x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_4:15
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) (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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (- 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) (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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) (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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) + x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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_4:16
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 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 ) & ( 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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) 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 ,) ^ : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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_4:17
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 (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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:18
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 (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 ) ) : ( ( ) ( ) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / ((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:19
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 (- (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 ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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
((- (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 ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / ((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:20
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) ) & 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 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) = 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:21
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 (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 ) ) : ( ( ) ( ) 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 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) & (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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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
((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 ) ) `| 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 ) ) = (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / (a : ( ( ) ( V22() V23() ext-real ) Real) + (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:22
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 (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 ) ) : ( ( ) ( ) 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) + x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) & 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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / ((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) : ( ( ) ( 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_4:23
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 (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 ) ) : ( ( ) ( ) 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) - a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) & 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) + a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 ) ) > 0 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) ^2) : ( ( ) ( 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 ) ) ) : ( ( ) ( 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_4:24
for a, b being ( ( ) ( V22() V23() ext-real ) Real)
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non 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 ) ) * (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 ) & ( 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) - a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) & 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) - b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 ) ) > 0 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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
((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 ) ) `| 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) - b : ( ( ) ( 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) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) * (x : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( 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_4:25
for a, b 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - b : ( ( ) ( 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 ) ) (#) 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 ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) & 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) - b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 ) ) > 0 : ( ( ) ( 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 ) ) ) & a : ( ( ) ( V22() V23() ext-real ) Real) - b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( 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 ) ) ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - b : ( ( ) ( 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 ) ) (#) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - b : ( ( ) ( 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 ) ) (#) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) - 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) - b : ( ( ) ( 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_4:26
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 (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 ) ) : ( ( ) ( ) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) - a : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( 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 : ( ( ) ( 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 ) ) ) & 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 : ( ( ) ( 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) <> 0 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) * (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
((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 ) ) `| 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) * (x : ( ( ) ( V22() V23() ext-real ) Real) - 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 ) ) ) ) ;

theorem :: FDIFF_4:27
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 ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * ((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 ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:28
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 ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:29
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 ((- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) (#) ((#R (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:30
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 (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:31
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 ((- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (- 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:32
for b, 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 ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) + (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & b : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( 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 ) ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 (3 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) + (b : ( ( ) ( 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 ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:33
for b, 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 ((- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) - (b : ( ( ) ( V22() V23() ext-real ) Real) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) & b : ( ( ) ( V22() V23() ext-real ) Real) <> 0 : ( ( ) ( 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 ) ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (- (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) : ( ( ) ( ) set ) ) ) * b : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) - (b : ( ( ) ( 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 ) ) #R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:34
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 ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) + 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 ) ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( (#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ^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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_4:35
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 (- ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) - 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 ) ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( - ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
((- ((#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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) ^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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_4:36
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 (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ,) + 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 ) ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) & 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 : ( ( ) ( 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 ) ) ) ) ) holds
( 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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) ) : ( ( ) ( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_4:37
for a, b 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 (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 ) ) : ( ( ) ( ) 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 ) ) = (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 ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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 ) ) * 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
((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 ) ) `| 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) * (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 ) ) . ((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 ) ) + b : ( ( ) ( 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_4:38
for a, b 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 (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 ) ) : ( ( ) ( ) 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 ) ) = (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 ) ) + b : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) 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 ) ) * 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
((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 ) ) `| 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) * (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 ) ) . ((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 ) ) + b : ( ( ) ( 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_4:39
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st ( 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( 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 ) ) ) ) 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 ) ) ^ : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) / ((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 ) ) ^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_4:40
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st ( 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) <> 0 : ( ( ) ( 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 ) ) ) ) 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 ) ) ^ : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) / ((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 ) ) ^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 ) ) ) ) ;

theorem :: FDIFF_4:41
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 ) ) (#) 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
( 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 ) ) (#) 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
((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 ) ) (#) 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 ) ) = cos (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:42
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 ) ) * 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 ) & ( 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( 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 ) ) ) ) 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 ) ) * 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
((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 ) ) * 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 ) ) = - (tan 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_4:43
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 ) ) * 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 ) ) : ( ( ) ( ) 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( 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 ) ) ) ) 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 ) ) * 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 ) ) 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 ) ) * 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 ) ) `| 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 ) ) = cot x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_4:44
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
( (- (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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
(((- (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) = (- (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 ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) * (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_4:45
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) : ( ( ) ( ) set ) holds
( (id 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 ) ) (#) 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 ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) `| 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 ) ) + (x : ( ( ) ( V22() V23() ext-real ) Real) * (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 ) ) ) ) ;

theorem :: FDIFF_4:46
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) + 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 ) ) : ( ( ) ( ) set ) holds
( ((- (id 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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) + 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 ) ) is_differentiable_on Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
((((- (id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) + 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 ) ) `| 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) * (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 ) ) ) ) ;

theorem :: FDIFF_4:47
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 ) ) + 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
( ((id 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 ) ) (#) 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 ) ) + 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
((((id 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 ) ) (#) 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 ) ) + 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 ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (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 ) ) ) ) ;

theorem :: FDIFF_4:48
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) : ( ( ) ( ) 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( 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 ) ) ) ) holds
( 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) `| 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 ) ) * ((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 ) ) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) ) ) ;

theorem :: FDIFF_4:49
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) : ( ( ) ( ) set ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) `| 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 ) ) * (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 ) ) ) ) ;

theorem :: FDIFF_4:50
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) & ( 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( 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 ) ) ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;

theorem :: FDIFF_4:51
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) - 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 ) & ( 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( 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 ) ) ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) - 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) - 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;

theorem :: FDIFF_4:52
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) & ( 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( 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 ) ) ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
( 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;

theorem :: FDIFF_4:53
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 ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) -valued V6() V39() V40() V41() ) Element of K19(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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) & ( 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 ) ) . x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V70() ) set ) ) > 0 : ( ( ) ( 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 ) ) ) & 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
( (- 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 ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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
(((- 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 ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) = ((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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;

theorem :: FDIFF_4:54
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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) : ( ( ) ( ) 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 : ( ( ) ( 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 ) ) ) holds
( (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * 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 ) ) `| 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 ) ) #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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) * (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 ) ) ) ) ;

theorem :: FDIFF_4:55
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) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) 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 ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) * (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_4:56
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 ) ) / (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 ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) / (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
((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 ) ) / (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 ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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_4:57
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 ) ) / 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 V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 : ( ( ) ( 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 ) ) ) ) ) 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 ) ) / 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
((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 ) ) / 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 ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive V49() V50() V51() V52() V53() V54() V68() V69() ) Element of NAT : ( ( ) ( non 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 ) ) ) ) ;