:: FDIFF_5 semantic presentation

begin

theorem :: FDIFF_5:1
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) 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() V62() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) 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() V62() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) <> 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:2
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) 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() V62() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) 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() V62() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) <> 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:3
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) 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() V62() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) 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() V62() ) set ) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) <> 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) / f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) 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() V62() ) 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() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:4
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = - (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:5
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:6
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:7
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:8
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) + ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:9
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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
( (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:10
for n being ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) holds
( (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:11
for n being ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) holds
( (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:12
for n being ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) holds
( (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = - ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:13
for n being ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) holds
( (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ((#Z n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * ((cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) #Z (n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V60() ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:14
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:15
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (- ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:16
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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) + (#R (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) + ((1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (x : ( ( ) ( V22() V23() ext-real ) Real) #R (- (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:17
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for g being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) holds
( g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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
((g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:18
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for g being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st not 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) holds
( g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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
((g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (cos : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) + (sin : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) . (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:19
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / x : ( ( ) ( V22() V23() ext-real ) Real) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:20
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = 1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) + (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:21
for Z being ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) )
for g being ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) st Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) c= dom (g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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
x : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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
((g : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = x : ( ( ) ( V22() V23() ext-real ) Real) + ((2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:22
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) > 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) / ((a : ( ( ) ( V22() V23() ext-real ) Real) - (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:23
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = a : ( ( ) ( V22() V23() ext-real ) Real) ) & f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V30( REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) , REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) > 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) > 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) * ((f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) + f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) / (f1 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) - f2 : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) PartFunc of ,) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = ((4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) * a : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * x : ( ( ) ( V22() V23() ext-real ) Real) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) / ((a : ( ( ) ( V22() V23() ext-real ) Real) |^ 2 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) - (x : ( ( ) ( V22() V23() ext-real ) Real) |^ 4 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:24
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) & ( for x being ( ( ) ( V22() V23() ext-real ) Real) st x : ( ( ) ( V22() V23() ext-real ) Real) in Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) holds
x : ( ( ) ( V22() V23() ext-real ) Real) > 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ((id Z : ( ( open ) ( open V49() V50() V51() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) (#) ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) * (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) - (ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;

theorem :: FDIFF_5:25
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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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
ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) <> 0 : ( ( ) ( V15() V16() V17() V21() V22() V23() ext-real non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V39() V40() V41() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) = - (1 : ( ( ) ( non empty V15() V16() V17() V21() V22() V23() ext-real positive non negative V49() V50() V51() V52() V53() V54() V60() V61() ) Element of NAT : ( ( ) ( non empty V15() V16() V17() V49() V50() V51() V52() V53() V54() V55() ) Element of K19(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V22() V23() ext-real ) Real) * ((ln : ( ( V6() ) ( Relation-like REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -defined REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) -valued V6() V39() V40() V41() ) Element of K19(K20(REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ,REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) 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() V62() ) set ) ) ^2) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) : ( ( ) ( V22() V23() ext-real ) Element of REAL : ( ( ) ( non empty V49() V50() V51() V55() V62() ) set ) ) ) ) ;