:: HFDIFF_1 semantic presentation

begin

theorem :: HFDIFF_1:1
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Function of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) holds dom (f : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Function of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) = Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:2
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) holds (- f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (- f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:3
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( dom ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) = REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) \ {0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V84() V85() V86() V87() V88() V89() ) set ) : ( ( ) ( V84() V85() V86() ) set ) & (#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) " {0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V84() V85() V86() V87() V88() V89() ) set ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) = {0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) ( V84() V85() V86() V87() V88() V89() ) set ) ) ;

theorem :: HFDIFF_1:4
for r, p being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds (r : ( ( ) ( V11() V12() ext-real ) Real) * p : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) (#) (p : ( ( ) ( V11() V12() ext-real ) Real) (#) ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:5
for f1 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,)
for n, m being ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) holds (n : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (m : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (n : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) + m : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:6
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:7
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) & f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:8
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds #Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ;

theorem :: HFDIFF_1:9
for x being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = - (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:10
for x being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = (- cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:11
for x being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:12
for x being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = - (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:13
for x being ( ( ) ( V11() V12() ext-real ) Real)
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:14
for x being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:15
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) + f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:16
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) - f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) - ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:17
for n, i being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) + f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:18
for n, i being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) <= n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) - f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) - ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:19
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) + f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:20
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) - f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:21
for r being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) (#) ((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:22
for r being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
r : ( ( ) ( V11() V12() ext-real ) Real) (#) f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:23
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:24
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) & f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:25
for x, r being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) * (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:26
for x, r being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) * (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:27
for x, r being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff ((r : ( ( ) ( V11() V12() ext-real ) Real) (#) exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = r : ( ( ) ( V11() V12() ext-real ) Real) * (exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:28
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:29
for x being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) <> 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V11() V12() ext-real ) Real) & diff (((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = - ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (x : ( ( ) ( V11() V12() ext-real ) Real) #Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / ((x : ( ( ) ( V11() V12() ext-real ) Real) #Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ;

theorem :: HFDIFF_1:30
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) (#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:31
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) >= 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) (#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:32
for n, m being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) > m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) (#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:33
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
( (diff ((- f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = - ((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) & - f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) ;

theorem :: HFDIFF_1:34
for x0, x being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x0 : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
( (Taylor (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x0 : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * ((x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) & (Taylor (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . (x0 : ( ( ) ( V11() V12() ext-real ) Real) + ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * ((x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ;

theorem :: HFDIFF_1:35
for r, x being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) st r : ( ( ) ( V11() V12() ext-real ) Real) > 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Maclaurin (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (x : ( ( ) ( V11() V12() ext-real ) Real) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) & (Maclaurin (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * PI : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (x : ( ( ) ( V11() V12() ext-real ) Real) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ;

theorem :: HFDIFF_1:36
for x being ( ( ) ( V11() V12() ext-real ) Real)
for n, m being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) > m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) & x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (x : ( ( ) ( V11() V12() ext-real ) Real) #Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:37
for x being ( ( ) ( V11() V12() ext-real ) Real)
for m being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff ((#Z m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ! : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:38
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds #Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:39
for x, a being ( ( ) ( V11() V12() ext-real ) Real)
for n, m being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) > m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((diff ((a : ( ( ) ( V11() V12() ext-real ) Real) (#) (#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = ((a : ( ( ) ( V11() V12() ext-real ) Real) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * (x : ( ( ) ( V11() V12() ext-real ) Real) #Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:40
for x, a being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff ((a : ( ( ) ( V11() V12() ext-real ) Real) (#) (#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = a : ( ( ) ( V11() V12() ext-real ) Real) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) !) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:41
for x0, x being ( ( ) ( V11() V12() ext-real ) Real)
for n, m being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st x0 : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) > m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Taylor ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = ((n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) choose m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (x0 : ( ( ) ( V11() V12() ext-real ) Real) #Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) - m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) * ((x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) |^ m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) & (Taylor ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ,x0 : ( ( ) ( V11() V12() ext-real ) Real) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = (x : ( ( ) ( V11() V12() ext-real ) Real) - x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ;

theorem :: HFDIFF_1:42
for n, m being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for r, x being ( ( ) ( V11() V12() ext-real ) Real) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) > m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) & r : ( ( ) ( V11() V12() ext-real ) Real) > 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Maclaurin ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . m : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) & (Maclaurin ((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,].(- r : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ,r : ( ( ) ( V11() V12() ext-real ) Real) .[ : ( ( ) ( open V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = x : ( ( ) ( V11() V12() ext-real ) Real) |^ n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ;

theorem :: HFDIFF_1:43
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:44
for x0 being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & x0 : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = - (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) / ((#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:45
for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) <> 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V11() V12() ext-real ) Real) & diff (((id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = - (1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V11() V12() ext-real ) Real) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ;

theorem :: HFDIFF_1:46
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((- 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:47
for x being ( ( ) ( V11() V12() ext-real ) Real) st x : ( ( ) ( V11() V12() ext-real ) Real) <> 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (#Z 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V11() V12() ext-real ) Real) & diff (((#Z 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = - ((2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) / ((x : ( ( ) ( V11() V12() ext-real ) Real) #Z 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ^2) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ;

theorem :: HFDIFF_1:48
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((#Z 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((- 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((#Z 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:49
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) holds
((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((- n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:50
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:51
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom ln : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom ((id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) holds
ln : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((id REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( total ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V22() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() V78() V80() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:52
for x0 being ( ( ) ( V11() V12() ext-real ) Real)
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) >= 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) & x0 : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & not 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
((diff (((#Z n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) = (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) * (((#Z (n : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) + 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) . x0 : ( ( ) ( V11() V12() ext-real ) Real) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ;

theorem :: HFDIFF_1:53
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (diff ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + ((- 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:54
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (diff ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + ((- 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:55
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (diff ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = 4 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (((- sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:56
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) holds
( tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: HFDIFF_1:57
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) holds
( cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: HFDIFF_1:58
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) holds
(diff (tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (((tan : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:59
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) holds
( cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((- 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: HFDIFF_1:60
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) holds
( sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^ : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (- ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: HFDIFF_1:61
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) c= dom cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V84() V85() V86() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) holds
(diff (cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (((cot : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:62
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (diff ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:63
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (diff ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (- sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:64
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = ((((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + ((3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (((diff (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (f1 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) ((diff (f2 : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:65
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (diff ((sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (- 8 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real V48() ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) (#) ((cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:66
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
(diff ((f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) ((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) ((f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ^2) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:67
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) )
for f being ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) st f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) is_differentiable_on 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) & ( for x0 being ( ( ) ( V11() V12() ext-real ) Real) st x0 : ( ( ) ( V11() V12() ext-real ) Real) in Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) . x0 : ( ( ) ( V11() V12() ext-real ) Real) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) <> 0 : ( ( ) ( V4() V5() V6() V10() V11() V12() ext-real non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
(diff ((f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ^) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (((2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) `| Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) - (((diff (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) / (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) (f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) (#) f : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) PartFunc of ,) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: HFDIFF_1:68
for Z being ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) holds (diff ((exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ,Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) )) : ( ( V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) ( V16() V19( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) V20( PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) V21() V30( NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) , PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ,(PFuncs (REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . 3 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) = (2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() ext-real positive non negative V48() V84() V85() V86() V87() V88() V89() V91() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V84() V85() V86() V87() V88() V89() V90() ) Element of K6(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( ) set ) ) ) (#) (exp_R : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) (#) ((- sin : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) + cos : ( ( V21() V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) ) ( non empty V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() total V30( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) , REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V21() ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) | Z : ( ( open ) ( open V84() V85() V86() ) Subset of ( ( ) ( ) set ) ) : ( ( ) ( V16() V19( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V20( REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) V21() V74() V75() V76() ) Element of K6(K7(REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ,REAL : ( ( ) ( non empty V57() V84() V85() V86() V90() ) set ) ) : ( ( ) ( V74() V75() V76() ) set ) ) : ( ( ) ( ) set ) ) ;